integral of (2022-cos(x))^5sin(x)

解

\int (2022 - cos (x))^{5} sin (x) d x

- 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) + 10221210 cos^{4} (x) - 2022 cos^{5} (x) + \frac{1}{6} cos^{6} (x) + C

解答ステップ

\int (2022 - cos (x))^{5} sin (x) d x

拡張

(2022 - cos (x))^{5} sin (x) : 202 2^{5} sin (x) - 202 2^{4} \cdot 5 sin (x) cos (x) + 202 2^{3} \cdot 10 cos^{2} (x) sin (x) - 40884840 cos^{3} (x) sin (x) + 10110 cos^{4} (x) sin (x) - cos^{5} (x) sin (x)

= \int 202 2^{5} sin (x) - 202 2^{4} \cdot 5 sin (x) cos (x) + 202 2^{3} \cdot 10 cos^{2} (x) sin (x) - 40884840 cos^{3} (x) sin (x) + 10110 cos^{4} (x) sin (x) - cos^{5} (x) sin (x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 202 2^{5} sin (x) d x - \int 202 2^{4} \cdot 5 sin (x) cos (x) d x + \int 202 2^{3} \cdot 10 cos^{2} (x) sin (x) d x - \int 40884840 cos^{3} (x) sin (x) d x + \int 10110 cos^{4} (x) sin (x) d x - \int cos^{5} (x) sin (x) d x

\int 202 2^{5} sin (x) d x = - 202 2^{5} cos (x)

\int 202 2^{4} \cdot 5 sin (x) cos (x) d x = 33 7^{4} \cdot 3240 sin^{2} (x)

\int 202 2^{3} \cdot 10 cos^{2} (x) sin (x) d x = - 27556382160 cos^{3} (x)

\int 40884840 cos^{3} (x) sin (x) d x = - 10221210 cos^{4} (x)

\int 10110 cos^{4} (x) sin (x) d x = - 2022 cos^{5} (x)

\int cos^{5} (x) sin (x) d x = - \frac{cos ^{6} ( x )}{6}

= - 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) - (- 10221210 cos^{4} (x)) - 2022 cos^{5} (x) - (- \frac{cos ^{6} ( x )}{6})

簡素化

- 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) - (- 10221210 cos^{4} (x)) - 2022 cos^{5} (x) - (- \frac{cos ^{6} ( x )}{6}) : - 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) + 10221210 cos^{4} (x) - 2022 cos^{5} (x) + \frac{1}{6} cos^{6} (x)

= - 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) + 10221210 cos^{4} (x) - 2022 cos^{5} (x) + \frac{1}{6} cos^{6} (x)

定数を解答に追加する

= - 202 2^{5} cos (x) - 33 7^{4} \cdot 3240 sin^{2} (x) - 27556382160 cos^{3} (x) + 10221210 cos^{4} (x) - 2022 cos^{5} (x) + \frac{1}{6} cos^{6} (x) + C