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受欢迎的微积分 >

integral from 1 to 2 of 1/(x(1+x^4))

解答

\int_{1}^{2} \frac{1}{x ( 1 + x ^{4} )} d x

解答

\frac{5 ln ( 2 ) - ln ( 17 )}{4}

+1

十进制

0.15813 \dots

求解步骤

\int_{1}^{2} \frac{1}{x ( 1 + x ^{4} )} d x

使用换元积分法

= \int_{2}^{17} \frac{1}{4 ( u - 1 ) u} d u

提出常数:

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

= \frac{1}{4} \cdot \int_{2}^{17} \frac{1}{( u - 1 ) u} d u

将

\frac{1}{( u - 1 ) u}

用部份分式展开:

\frac{1}{u - 1} - \frac{1}{u}

= \frac{1}{4} \cdot \int_{2}^{17} \frac{1}{u - 1} - \frac{1}{u} 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

= \frac{1}{4} (\int_{2}^{17} \frac{1}{u - 1} d u - \int_{2}^{17} \frac{1}{u} d u)

\int_{2}^{17} \frac{1}{u - 1} d u = 4 ln (2)

\int_{2}^{17} \frac{1}{u} d u = ln (17) - ln (2)

= \frac{1}{4} (4 ln (2) - (ln (17) - ln (2)))

化简

\frac{1}{4} (4 ln (2) - (ln (17) - ln (2))) : \frac{5 ln ( 2 ) - ln ( 17 )}{4}

= \frac{5 ln ( 2 ) - ln ( 17 )}{4}

作图

流行的例子

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\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 cos^{4} (x) d x

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\int_{0}^{\frac{π}{3}} sec (θ) tan (θ) d θ

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\int_{0}^{2 π} 1 + x^{2} d x

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