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受欢迎的微积分 >

integral of 2sin^3(x)cos^4(x)

解答

\int 2 sin^{3} (x) cos^{4} (x) d x

解答

2 (- \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7}) + C

求解步骤

\int 2 sin^{3} (x) cos^{4} (x) d x

提出常数:

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

= 2 \cdot \int sin^{3} (x) cos^{4} (x) d x

使用三角恒等式改写

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

使用换元积分法

= 2 \cdot \int - u^{4} (1 - u^{2}) d u

乘开

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

= 2 \cdot \int - u^{4} + u^{6} d u

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

= 2 (- \int u^{4} d u + \int u^{6} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

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

= 2 (- \frac{u ^{5}}{5} + \frac{u ^{7}}{7})

u = cos (x)

代回

= 2 (- \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7})

解答补常数

= 2 (- \frac{cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7}) + C

作图

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\int sin (\frac{1}{5} x) d x

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