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受欢迎的微积分 >

integral of sec^4(3x)tan^4(3x)

解答

\int sec^{4} (3 x) tan^{4} (3 x) d x

解答

\frac{1}{3} (\frac{tan ^{5} ( 3 x )}{5} + \frac{tan ^{7} ( 3 x )}{7}) + C

求解步骤

\int sec^{4} (3 x) tan^{4} (3 x) d x

使用换元积分法

= \int sec^{4} (u) tan^{4} (u) \frac{1}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int sec^{4} (u) tan^{4} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int (1 + tan^{2} (u)) sec^{2} (u) tan^{4} (u) d u

使用换元积分法

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

乘开

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

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

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

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

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

代回

= \frac{1}{3} (\frac{tan ^{5} ( 3 x )}{5} + \frac{tan ^{7} ( 3 x )}{7})

解答补常数

= \frac{1}{3} (\frac{tan ^{5} ( 3 x )}{5} + \frac{tan ^{7} ( 3 x )}{7}) + C

作图

流行的例子

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(dy}{dx}=\frac{y+3)/x

\frac{d y}{d x} = \frac{y + 3}{x}

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