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受欢迎的微积分 >

integral of (csc^4(x))/(cot^6(x))

解答

\int \frac{csc ^{4} ( x )}{cot ^{6} ( x )} d x

解答

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

求解步骤

\int \frac{csc ^{4} ( x )}{cot ^{6} ( x )} d x

使用三角恒等式改写

= \int \frac{tan ^{2} ( x )}{cos ^{4} ( x )} d x

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{4} (x) tan^{2} (x) d x

使用三角恒等式改写

= \int (1 + tan^{2} (x)) sec^{2} (x) tan^{2} (x) d x

使用换元积分法

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

乘开

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

= \int u^{2} + u^{4} 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^{2} d u + \int u^{4} d u

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

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

= \frac{u ^{3}}{3} + \frac{u ^{5}}{5}

u = tan (x)

代回

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

解答补常数

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

作图

流行的例子

integral of 1/((sqrt(x^2+1)))

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

integral of (e^{3x}-e^{2x})/(e^x)

\int \frac{e ^{3 x} - e ^{2 x}}{e ^{x}} d x

integral of [u^5-u^3+u^2]

\int [u^{5} - u^{3} + u^{2}] d u

integral of x/(1+4x)

\int \frac{x}{1 + 4 x} d x

integral of 1/(sqrt(x)(3+\sqrt{x))}

\int \frac{1}{x ( 3 + x )} d x