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受欢迎的微积分 >

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

解答

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

解答

- cos^{4} (x) + \frac{2}{3} cos^{6} (x) + C

求解步骤

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

- u^{3} (1 - u^{2}) : - u^{3} + u^{5}

= 4 \cdot \int - u^{3} + u^{5} 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

= 4 (- \int u^{3} d u + \int u^{5} d u)

\int u^{3} d u = \frac{u ^{4}}{4}

\int u^{5} d u = \frac{u ^{6}}{6}

= 4 (- \frac{u ^{4}}{4} + \frac{u ^{6}}{6})

u = cos (x)

代回

= 4 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6})

化简

4 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6}) : - cos^{4} (x) + \frac{2}{3} cos^{6} (x)

= - cos^{4} (x) + \frac{2}{3} cos^{6} (x)

解答补常数

= - cos^{4} (x) + \frac{2}{3} cos^{6} (x) + C

作图

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