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受欢迎的微积分 >

integral of 4cos^3(5x)

解答

\int 4 cos^{3} (5 x) d x

解答

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

求解步骤

\int 4 cos^{3} (5 x) d x

提出常数:

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

= 4 \cdot \int cos^{3} (5 x) d x

使用换元积分法

= 4 \cdot \int cos^{3} (u) \frac{1}{5} d u

提出常数:

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

= 4 \cdot \frac{1}{5} \cdot \int cos^{3} (u) d u

使用三角恒等式改写

= 4 \cdot \frac{1}{5} \cdot \int (1 - sin^{2} (u)) cos (u) d u

使用换元积分法

= 4 \cdot \frac{1}{5} \cdot \int 1 - v^{2} 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

= 4 \cdot \frac{1}{5} (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

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

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

代回

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

化简

4 \cdot \frac{1}{5} (sin (5 x) - \frac{sin ^{3} ( 5 x )}{3}) : \frac{4}{5} (sin (5 x) - \frac{1}{3} sin^{3} (5 x))

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

解答补常数

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

作图

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