Problem
Mühendislik ve Doğa Bilimleri Fakūltesi 2019-2020 Bahar Dōnemi Genel Matematik II Vize Smav Ödev Sorular 04-12 Mays 2020 Adi - Soyadı: Numarası : Bōlümũ : 1) \( \int \frac{\sin x}{\cos ^{7} x} d x \) integralini hesaplaymuz. 2) \( \int_{0}^{1} \frac{x \arctan \left(x^{2}\right)}{1+x^{4}} d x \quad \) integralini hesaplayiniz. 3) \( y=x^{2} \) ile \( y=8-x^{2} \) eğrileri tarafindan smurlanan bōlgenin alanum bulunuz 4) \( \int \cos ^{6} x \sin x \tan ^{2} x d x \) integralini hesaplaymiz. 5) \( \int \frac{2 x+5}{x^{3}+5 x} d x \) integralini hesaplayınz. 6) \( f(x)=\int_{1}^{t} \sin (\ln t) d t \) ve \( g(x)=\int_{1}^{e^{t}} \cos (\ln t) d t \) olmak ūzere \( f\left(\frac{\pi}{2}\right)+g\left(\frac{\pi}{2}\right) \) toplaminı bulunuz 7) \( \int_{-\pi}^{\pi}\left|\cos ^{3} x\right| d x \) integralini hesaplayınz 8) \( \int \frac{d x}{\sqrt{-x^{2}+4 x+5}} \) integralini hesaplayinı.
Solution
Step 1 ：\(\int \frac{\sin x}{\cos ^{7} x} d x = \int \sin x \cos^{-7}x dx\)
Step 2 ：\(\int_{0}^{1} \frac{x \arctan \left(x^{2}\right)}{1+x^{4}} d x\)
Step 3 ：\(\int \cos ^{6} x \sin x \tan ^{2} x d x \)
Step 4 ：\(\int \frac{2 x+5}{x^{3}+5 x} d x\)
Step 5 ：\(f(x)=\int_{1}^{t} \sin (\ln t) d t\); \(g(x)=\int_{1}^{e^{t}} \cos (\ln t) d t\); \(f\left(\frac{\pi}{2}\right)+g\left(\frac{\pi}{2}\right)\)
Step 6 ：\(\int_{-\pi}^{\pi}\left|\cos ^{3} x\right| d x\)
Step 7 ：\(\int \frac{d x}{\sqrt{-x^{2}+4 x+5}}\)
