integral of (cos^4(x))/(sin^2(x))

解

\int \frac{cos ^{4} ( x )}{sin ^{2} ( x )} d x

- cos^{4} (x) cot (x) - 4 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) + C

解答ステップ

\int \frac{cos ^{4} ( x )}{sin ^{2} ( x )} d x

部分積分を適用する

= - cos^{4} (x) cot (x) - \int 4 cos^{3} (x) sin (x) cot (x) d x

\int 4 cos^{3} (x) sin (x) cot (x) d x = 4 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

= - cos^{4} (x) cot (x) - 4 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

定数を解答に追加する

= - cos^{4} (x) cot (x) - 4 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) + C