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受欢迎的微积分 >

integral of cot^3(a)csc^3(a)

解答

\int cot^{3} (a) csc^{3} (a) d a

解答

- \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a) + C

求解步骤

\int cot^{3} (a) csc^{3} (a) d a

使用换元积分法

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

提出常数:

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

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

乘开

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

= - \int u^{4} - u^{2} 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

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

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

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

= - (\frac{u ^{5}}{5} - \frac{u ^{3}}{3})

u = csc (a)

代回

= - (\frac{csc ^{5} ( a )}{5} - \frac{csc ^{3} ( a )}{3})

化简

- (\frac{csc ^{5} ( a )}{5} - \frac{csc ^{3} ( a )}{3}) : - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a)

= - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a)

解答补常数

= - \frac{1}{5} csc^{5} (a) + \frac{1}{3} csc^{3} (a) + C

作图

流行的例子

in t e g r a l e^{7 x}

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\int \frac{5 x}{1 + x y} d y

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\int 2^{x^{2} + 1} x d x

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\int \frac{x ^{3} - 6 x - 4}{x + 2} d x

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\int \frac{4 y ^{2} - 6 y - 12}{y ( y + 2 ) ( y - 3 )} d y