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Eğitmen önce trigonometrik eşitlikleri ve toplam-fark formüllerini hatırlatarak başlıyor, ardından sin²x, cos²x, tanx ve secx fonksiyonlarının integral hesaplamalarını adım adım gösteriyor. Değişken değiştirme ve kısmi integrasyon teknikleri kullanılarak çeşitli trigonometrik integral soruları çözülmektedir.","Video boyunca beş farklı soru çözülmekte ve her soru için detaylı çözüm süreci gösterilmektedir. Özellikle sinüs ve kosinüs fonksiyonlarının çift kuvvetlerinin integral hesaplamalarında dikkat edilmesi gereken noktalar vurgulanmakta, tanjant ve sekant fonksiyonlarının temel özellikleri hatırlatılmakta ve belirli integralde kullanılan dönüşüm teknikleri örneklerle açıklanmaktadır."]},"endTime":2030,"title":"Trigonometrik İntegraller ve Dönüşümler Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Trigonometrik İntegraller ve Dönüşümler","list":{"type":"unordered","items":["Trigonometrik integraller ve dönüşümler, belirsiz integralin son konusudur.","Değişken değiştirme konusu daha önce defalarca ele alınmış olup, bu derste tanjant dönüşümleri daha önemli olacaktır.","Trigonometrik integraller konusunu tek bir başlık altında anlatmak istenmektedir."]},"beginTime":0,"endTime":35,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=0&ask_summarization=1"},{"index":1,"title":"Trigonometrik Eşitlikler","list":{"type":"unordered","items":["Trigonometrik eşitlikler arasında sin²x + cos²x = 1 ilişkisi en önemlidir.","Sinüs ve kosinüs toplam-fark formülleri hatırlatılmaktadır.","Kosinüs 2x ifadesi cos²x - sin²x veya 1 - 2sin²x olarak yazılabilir."]},"beginTime":35,"endTime":124,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=35&ask_summarization=1"},{"index":2,"title":"Trigonometrik Dönüşümler","list":{"type":"unordered","items":["sinx = (1 - cos2x) / 2 şeklinde yazılabilir, böylece sinx'in integrali alınabilir.","cos2x = (cos²x + 1) / 2 şeklinde de yazılabilir.","Doğrudan sinx veya cos²x gibi ifadeler yerine bu dönüşümleri kullanarak integral alınabilir."]},"beginTime":124,"endTime":211,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=124&ask_summarization=1"},{"index":3,"title":"Örnek Uygulama","list":{"type":"unordered","items":["sin5x dx integralinde sinx'e u değişkeni atanabilir.","sin5x dx ifadesi sin4x sinx dx şeklinde parçalanabilir.","sinx = √(1 - cos²x) dönüşümü yapılarak integral alınabilir."]},"beginTime":211,"endTime":278,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=211&ask_summarization=1"},{"index":4,"title":"İntegral Hesaplama Yöntemleri","list":{"type":"unordered","items":["Kosinüs x'e u diyerek türev alındığında eksi sinüs x dx elde edilir, bu da sinüs x dx'in u'nun tersi olduğunu gösterir.","İntegral hesaplamasında değişken değiştirme yöntemi kullanılarak, sinüs x'e u diyerek integral daha kolay çözülebilir.","İntegral hesaplamasında, cosinüs x dx'in kalması için sinüs x'e u diyerek integral daha basit hale getirilebilir."]},"beginTime":280,"endTime":527,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=280&ask_summarization=1"},{"index":5,"title":"Sinüs ve Kosinüs Dönüşümleri","list":{"type":"unordered","items":["Sinüs x artı sinüs x ifadesi sinüs 2x'e eşittir ve sinüs x çarpı cosinüs x'in karesine eşittir.","Sinüs 2x'in karesi integralinde, 2x'e u diyerek integral daha kolay çözülebilir.","Sinüs 2x'in karesi integralinde, sinüs 2x'in karesi ifadesi 1/2 cos 2x'e eşittir."]},"beginTime":527,"endTime":724,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=527&ask_summarization=1"},{"index":6,"title":"Genel İntegral Çözüm Yöntemleri","list":{"type":"unordered","items":["Cosinüs 2x'in karesi, cos 2x artı 1 bölü 2'nin karesi olarak ifade edilebilir.","İntegral hesaplamasında, cosinüs 2x'in karesi ifadesi 1/4 cos 4x'e eşittir.","İntegral hesaplamasında, cosinüs 2x'in karesi ifadesi 1/4 sin 4x'e eşittir."]},"beginTime":724,"endTime":876,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=724&ask_summarization=1"},{"index":7,"title":"Trigonometrik Fonksiyonların İntegral Özellikleri","list":{"type":"unordered","items":["Trigonometrik fonksiyonların integral özellikleri sadece sinüs ve kosinüs değil, tanjant ve sekant fonksiyonları için de geçerlidir.","Tanjant x'in türevi secant x'tir ve secant x'in türevi tanjant x'tir.","Hiperbolik fonksiyonlar ve kotanjant, kosekant fonksiyonları için de benzer eşitlikler geçerlidir."]},"beginTime":877,"endTime":947,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=877&ask_summarization=1"},{"index":8,"title":"Tanjant Fonksiyonunun İntegrali","list":{"type":"unordered","items":["İntegral tanjant 8x dx integrali, tanjant x'in türevinin secant x olduğunu kullanarak çözülebilir.","Değişken değiştirme yöntemi kullanılarak integral tanjant 8x dx = tanjant 11x/11 + tanjant 9x/9 + C şeklinde çözülebilir.","Tanjant fonksiyonlarının integrali sinüs ve kosinüs fonksiyonlarına göre daha kolay ayırt edilebilir."]},"beginTime":947,"endTime":1034,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=947&ask_summarization=1"},{"index":9,"title":"Sekant Fonksiyonunun İntegrali","list":{"type":"unordered","items":["İntegral secant x dx integrali, secant x'i u olarak değişken değiştirerek çözülebilir.","İntegral secant x dx = 2ln|secant x| + C şeklinde çözülebilir.","Sekant fonksiyonunun integrali, tanjant fonksiyonunun integraliyle ilişkilidir."]},"beginTime":1034,"endTime":1159,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1034&ask_summarization=1"},{"index":10,"title":"Sekant ve Tanjant Fonksiyonlarının İntegrali","list":{"type":"unordered","items":["İntegral secant x tanjant x dx integrali, kısmi integrasyon ve değişken değiştirme yöntemleri kullanılarak çözülebilir.","İntegral secant x tanjant x dx = ln|secant x + tanjant x| + C şeklinde çözülebilir.","Bu integral, sekant fonksiyonunun integraliyle ilişkilidir ve ln|secant x + tanjant x| şeklinde ifade edilir."]},"beginTime":1159,"endTime":1407,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1159&ask_summarization=1"},{"index":11,"title":"Trigonometrik Dönüşüm Formülleri","list":{"type":"unordered","items":["Sinüs ve kosinüs fonksiyonlarının çarpım formülleri kullanılarak sinüs a çarpı sinüs b ifadesi 1/2 sinüs (a+b) + 1/2 sinüs (a-b) olarak hesaplanabilir.","Kosinüs fonksiyonlarının toplamı ve farkı formülleri de benzer şekilde kullanılabilir.","Sinüs tek fonksiyon olduğu için sinüs (-x) = -sinüs x, kosinüs ise çift fonksiyon olduğu için kosinüs (-x) = kosinüs x olarak hesaplanır."]},"beginTime":1412,"endTime":1594,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1412&ask_summarization=1"},{"index":12,"title":"Kök İçindeki İfadelerin Dönüşümü","list":{"type":"unordered","items":["Kök içerisinde a² - x² ifadesi varsa, x = a sin teta dönüşümü uygulanabilir ve kök içindeki ifade |a cos teta| olarak dışarı çıkar.","Kök içerisinde a² + x² ifadesi varsa, x = a sec teta dönüşümü uygulanabilir ve kök içindeki ifade |sec teta - 1| olarak dışarı çıkar.","Bu dönüşümler belirli integral hesaplamalarında kullanılır ve uygun tanım aralıklarında geçerlidir."]},"beginTime":1594,"endTime":1743,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1594&ask_summarization=1"},{"index":13,"title":"Dik Üçgen ve Trigonometrik Dönüşümler","list":{"type":"unordered","items":["x = a sin teta dönüşümü uygulandığında, x/a = sin teta olur ve bu durumda bir dik üçgen oluşur.","Dik üçgende taban a² - x², komşu kenar x, hipotenüs a olarak hesaplanır.","x = sin u dönüşümü uygulandığında, u = sin x olur ve bu durumda da bir dik üçgen oluşur."]},"beginTime":1743,"endTime":1881,"href":"/video/preview/9485995633585533511?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1743&ask_summarization=1"},{"index":14,"title":"Sekant ve Tanjant Dönüşümleri","list":{"type":"unordered","items":["x² - 9 ifadesi varsa, x = 3 sec u dönüşümü uygulanabilir ve bu durumda x/3 = sec u olur.","Sekant fonksiyonu 1/cos u olarak tanımlanır ve bu dönüşümde trigonometrik ilişkiler kullanılarak integral hesaplanabilir.","Tanjant dönüşümü ve tan(x/2) dönüşümü de belirsiz integral hesaplamalarında 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Eğitmen, parametrize etme işlemi yaparak vektör alanının çizgi integralini hesaplama sürecini iki farklı örnek üzerinden göstermektedir. Video, vektör alanının çizgi integralinin nasıl hesaplanacağını gösteren bir örnekle sona ererken, bir sonraki videoda vektör alanı konsorveti (korumacı vektör alanı) konusunun ele alınacağı belirtilmektedir.","Eğitmen, bu konuyla ilgili örneklerden oluşan beş serilik bir video serisi hazırlayacağını da eklemektedir."]},"endTime":1302,"title":"Vektör Alanlarının Çizgi İntegrali Hesaplama Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Vektör Alanlarının Çizgi İntegrali","list":{"type":"unordered","items":["Bu videoda vektör alanlarının çizgi integralinin nasıl hesaplanacağı anlatılacaktır.","Vektör alanı, koordinatları değişkenlerden oluşan vektörlere verilen genel isimdir ve büyük F harfi ile gösterilir.","Vektör alanı iki değişkenli (p(x,y), q(x,y)) veya üç değişkenli (p(x,y,z), q(x,y,z), r(x,y,z)) 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Örnek soru üzerinden dikdörtgensel ve dikdörtgensel olmayan bölgelerin sınır belirleme ve yerleştirme işlemleri gösterilmektedir.","Video, integral sorusunun çözümünün son adımıyla başlayıp, değişken değiştirme yöntemiyle çift katlı integralin çözüm adımlarını gözden geçirmektedir."]},"endTime":1157,"title":"Çift Katlı İntegralde Değişken Değiştirme Yöntemi","beginTime":0}],"fullResult":[{"index":0,"title":"Çift Katlı İntegralde Değişken Değiştirme Örneği","list":{"type":"unordered","items":["Video, çift katlı integrallerde değişken değiştirme konusuna örnek bir soru sunuyor.","Soruda xyda integrali, x=2u+3 ve y=2u-3 dönüşümleri kullanılarak hesaplanacak.","R bölgesi, köşe koordinatları (0,0), (5,0), (5,2) ve (5,-2) olan dörtgensel bir alan."]},"beginTime":1,"endTime":38,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1&ask_summarization=1"},{"index":1,"title":"Jakobi Yan 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oluşturulacaktır."]},"beginTime":159,"endTime":287,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=159&ask_summarization=1"},{"index":3,"title":"Sınır Fonksiyonlarının Bulunması","list":{"type":"unordered","items":["R bölgesinin sınırları, x=2u+3 ve y=2u-3 dönüşümleri kullanılarak u ve v koordinatlarına çevrilir.","Her sınır fonksiyonu için x ve y yerine dönüşüm fonksiyonları konularak yeni denklemler elde edilir.","Tüm sınır fonksiyonları u ve v koordinatlarında bulunur ve dikdörtgensel bir bölge oluşturur."]},"beginTime":287,"endTime":836,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=287&ask_summarization=1"},{"index":4,"title":"İntegralin Hesaplanması","list":{"type":"unordered","items":["Üçüncü adım olarak, integral sınırları belirlenir: u'nun sınırları 0'dan 5/4'e, v'nin sınırları 0'dan 5/6'ya kadar.","İntegral, dışa v'nin sınırları, içe u'nun sınırları şeklinde yazılır.","İntegral hesaplanarak sonuç 75/2 bulunur."]},"beginTime":836,"endTime":1009,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=836&ask_summarization=1"},{"index":5,"title":"İntegral Hesaplama","list":{"type":"unordered","items":["Son adımda v'ye göre integral alınarak 75v²/5/6 sonucu elde edildi.","İntegral hesaplaması sonucunda 25/225/4 olarak bulundu."]},"beginTime":1018,"endTime":1061,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1018&ask_summarization=1"},{"index":6,"title":"Çift Katlı İntegral Çözüm Adımları","list":{"type":"unordered","items":["Değişken değiştirme yaparak çift katlı integralin çözüm adımları gözden geçirildi.","Çözüm adımları: 1) Jakobiyan hesaplaması, 2) R bölgesinin çizimi, 3) Fonksiyonların içine dönüşüm yapılması.","Dönüşüm sonrası elde edilen fonksiyonların çizimi yapılarak dikdörtgensel bir bölge elde edildi."]},"beginTime":1061,"endTime":1109,"href":"/video/preview/5770806488750006446?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1061&ask_summarization=1"},{"index":7,"title":"İntegral Sınırları ve Yerleştirme","list":{"type":"unordered","items":["Çift katlı integralde sınırlar dikdörtgensel bölgeler ve dikdörtgensel olmayan bölgeler olarak ikiye ayrılır.","Dikdörtgensel bölgelerde iki sınırda sayı, dikdörtgensel olmayanlarda bir sınır sayı bir sınır fonksiyondur.","Yerleştirme kısmında önce u sonra v yerleştirilir, iki sınırda sayı ise keyfinize göre davranabilirsiniz.","Yerleştirme sırasında jakobiyan mutlak değer içinde yazılır ve x-y yerlerine dönüşümler 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Eğitmen, çok değişkenli fonksiyonlarda yerel maksimum, minimum ve eğer noktalarını bulma konusunu örnek üzerinden açıklamaktadır.","Video, önce konu anlatımının yapıldığını belirterek başlıyor ve ardından sınavlarda sorulabilecek kalitede bir örnek üzerinden çözüm sürecini adım adım gösteriyor. Örnek, f(x,y) = 9x³ + 4xy fonksiyonunun yerel maksimum, minimum ve eğer noktalarını bulmayı içeriyor. Çözüm üç adımda ilerliyor: ilk olarak gradient'i sıfıra eşitleyerek kritik noktaları bulma, ikinci olarak d değerini hesaplama ve son olarak bu noktaları sınıflandırma. Video, sınavlarda bu konuyu öğrenmek isteyenler için faydalı bir kaynak niteliğindedir."]},"endTime":618,"title":"Çok Değişkenli Fonksiyonlarda Yerel Maksimum, Minimum ve Eger Noktası Bulma","beginTime":0}],"fullResult":[{"index":0,"title":"Giriş ve Soru Tanıtımı","list":{"type":"unordered","items":["Video, yerel maksimum, yerel minimum ve eğer noktası bulma konusunu ele alıyor.","Konu anlatım videosunu izlememiş kişilerin önce o videoyu izlemesi öneriliyor.","Soru, f(x,y) = 9x³ + y⁴ - 4xy fonksiyonunun yerel maksimum, yerel minimum ve eğer noktalarını bulmayı gerektiriyor."]},"beginTime":1,"endTime":54,"href":"/video/preview/12476430028588655811?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=1&ask_summarization=1"},{"index":1,"title":"Kritik Noktaları Bulma","list":{"type":"unordered","items":["Çok değişkenli fonksiyonlarda yerel maksimum, yerel minimum ve eğer noktalarını bulmak için ilk adım gradient'in sıfıra eşitlenmesidir.","Gradient, f fonksiyonunun x'e ve y'ye göre kısmi türevlerinden oluşur.","Kritik noktalar, fx ve fy türevlerini sıfıra eşitlerken bulunan (x,y) çiftleridir."]},"beginTime":54,"endTime":143,"href":"/video/preview/12476430028588655811?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=54&ask_summarization=1"},{"index":2,"title":"Denklemlerin Çözümü","list":{"type":"unordered","items":["Denklemlerin birinde x veya y'yi yalnız bırakıp diğer denklemde yerine koyarak nokta çiftleri bulunur.","Çözüm sürecinde x = 0 ve x = 4/9 değerleri bulunur.","Bu değerler kullanılarak (0,0) ve (4/9, 4/3) kritik noktaları elde edilir."]},"beginTime":143,"endTime":377,"href":"/video/preview/12476430028588655811?parent-reqid=1769428838450002-10512957915936845195-balancer-l7leveler-kubr-yp-sas-25-BAL&text=Mr+Calculus&t=143&ask_summarization=1"},{"index":3,"title":"Sınıflama ve Sonuç","list":{"type":"unordered","items":["Kritik noktaları 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