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受欢迎的微积分 >

integral of xtan^4(x^2)

解答

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

解答

\frac{1}{2} (\frac{1}{3} tan^{3} (x^{2}) + x^{2} - tan (x^{2})) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用降次积分法

= \frac{1}{2} (\frac{tan ^{3} ( u )}{3} - \int tan^{2} (u) d u)

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

= \frac{1}{2} (\frac{tan ^{3} ( u )}{3} - (- u + tan (u)))

u = x^{2}

代回

= \frac{1}{2} (\frac{tan ^{3} ( x ^{2} )}{3} - (- x^{2} + tan (x^{2})))

化简

\frac{1}{2} (\frac{tan ^{3} ( x ^{2} )}{3} - (- x^{2} + tan (x^{2}))) : \frac{1}{2} (\frac{1}{3} tan^{3} (x^{2}) + x^{2} - tan (x^{2}))

= \frac{1}{2} (\frac{1}{3} tan^{3} (x^{2}) + x^{2} - tan (x^{2}))

解答补常数

= \frac{1}{2} (\frac{1}{3} tan^{3} (x^{2}) + x^{2} - tan (x^{2})) + C

作图

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