integral of (x^3csc^2(x^4))/(cot(x^4))

解答

\int \frac{x ^{3} csc ^{2} ( x ^{4} )}{cot ( x ^{4} )} d x

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

求解步骤

\int \frac{x ^{3} csc ^{2} ( x ^{4} )}{cot ( x ^{4} )} d x

使用三角恒等式改写

= \int \frac{x ^{3} ( 1 + cot ^{2} ( x ^{4} ) )}{cot ( x ^{4} )} d x

乘开

\frac{x ^{3} ( 1 + cot ^{2} ( x ^{4} ) )}{cot ( x ^{4} )} : \frac{x ^{3}}{cot ( x ^{4} )} + x^{3} cot (x^{4})

= \int \frac{x ^{3}}{cot ( x ^{4} )} + x^{3} cot (x^{4}) d x

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 \frac{x ^{3}}{cot ( x ^{4} )} d x + \int x^{3} cot (x^{4}) d x

\int \frac{x ^{3}}{cot ( x ^{4} )} d x = - \frac{1}{4} ln cos (x^{4})

\int x^{3} cot (x^{4}) d x = \frac{1}{4} ln sin (x^{4})

= - \frac{1}{4} ln cos (x^{4}) + \frac{1}{4} ln sin (x^{4})

解答补常数

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

\int (4 x - y) d x

\int \frac{( x - 1 ) ^{2}}{x ^{2}} d x

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\int sin (15 x) sin (16 x) d x