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受欢迎的微积分 >

integral of (cos(ax))/(sin^5(ax))

解答

\int \frac{cos ( a x )}{sin ^{5} ( a x )} d x

解答

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

求解步骤

\int \frac{cos ( a x )}{sin ^{5} ( a x )} d x

使用三角恒等式改写

= \int \frac{cot ( a x )}{sin ^{4} ( a x )} d x

使用基本三角恒等式:

\frac{1}{sin ( x )} = csc (x)

= \int csc^{4} (a x) cot (a x) d x

使用换元积分法

= \int \frac{csc ^{4} ( u ) cot ( u )}{a} d u

提出常数:

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

= \frac{1}{a} \cdot \int csc^{4} (u) cot (u) d u

使用三角恒等式改写

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

使用换元积分法

= \frac{1}{a} \cdot \int \frac{1}{v ^{5}} d v

使用幂法则

= \frac{1}{a} (- \frac{1}{4 v ^{4}})

代回

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

化简

\frac{1}{a} (- \frac{1}{4 sin ^{4} ( a x )}) : - \frac{1}{4 a sin ^{4} ( a x )}

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

解答补常数

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

作图

流行的例子

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\frac{d}{d x} (x y e^{\frac{- x ^{2}}{2} - \frac{y ^{2}}{2}})

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x^{^{'}} + (2 sin (t) cos (t)) x = e^{- s i n^{2} (t)}

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\int ln (r) d r

y^{^{'}} = - 6 x y^{4}

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\frac{d}{d x} (\frac{5 x ^{2} - x}{x})