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受欢迎的微积分 >

integral of cos^7(4x)

解答

\int cos^{7} (4 x) d x

解答

\frac{1}{4} (sin (4 x) - sin^{3} (4 x) + \frac{3 sin ^{5} ( 4 x )}{5} - \frac{sin ^{7} ( 4 x )}{7}) + C

求解步骤

\int cos^{7} (4 x) d x

使用换元积分法

= \int cos^{7} (u) \frac{1}{4} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int cos^{7} (u) d u

使用三角恒等式改写

= \frac{1}{4} \cdot \int (1 - sin^{2} (u))^{3} cos (u) d u

使用换元积分法

= \frac{1}{4} \cdot \int (1 - v^{2})^{3} d v

乘开

(1 - v^{2})^{3} : 1 - 3 v^{2} + 3 v^{4} - v^{6}

= \frac{1}{4} \cdot \int 1 - 3 v^{2} + 3 v^{4} - v^{6} d v

使用积分加法定则:

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

= \frac{1}{4} (\int 1 d v - \int 3 v^{2} d v + \int 3 v^{4} d v - \int v^{6} d v)

\int 1 d v = v

\int 3 v^{2} d v = v^{3}

\int 3 v^{4} d v = \frac{3 v ^{5}}{5}

\int v^{6} d v = \frac{v ^{7}}{7}

= \frac{1}{4} (v - v^{3} + \frac{3 v ^{5}}{5} - \frac{v ^{7}}{7})

代回

= \frac{1}{4} (sin (4 x) - sin^{3} (4 x) + \frac{3 sin ^{5} ( 4 x )}{5} - \frac{sin ^{7} ( 4 x )}{7})

解答补常数

= \frac{1}{4} (sin (4 x) - sin^{3} (4 x) + \frac{3 sin ^{5} ( 4 x )}{5} - \frac{sin ^{7} ( 4 x )}{7}) + C

作图

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