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integral of (sqrt(36+x^2))/x

解答

\int \frac{36 + x ^{2}}{x} d x

解答

- 3 ln \frac{1}{6} x^{2} + 36 + 1 + 3 ln \frac{1}{6} x^{2} + 36 - 1 + 36 + x^{2} + C

求解步骤

\int \frac{36 + x ^{2}}{x} d x

使用换元积分法

= \int \frac{u ^{2}}{u ^{2} - 36} d u

因式分解

u^{2} - 36 : - (- u^{2} + 36)

= \int \frac{u ^{2}}{- ( - u ^{2} + 36 )} d u

提出常数:

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

= - \int \frac{u ^{2}}{- u ^{2} + 36} d u

使用换元积分法

= - \int \frac{6 v ^{2}}{- v ^{2} + 1} d v

提出常数:

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

= - 6 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 6 \cdot \int \frac{1}{- v ^{2} + 1} - 1 d v

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

= - 6 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v)

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 6 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

代回

= - 6 \frac{ln \frac{36 + x ^{2}}{6} + 1}{2} - \frac{ln \frac{36 + x ^{2}}{6} - 1}{2} - \frac{36 + x ^{2}}{6}

化简

- 6 \frac{ln \frac{36 + x ^{2}}{6} + 1}{2} - \frac{ln \frac{36 + x ^{2}}{6} - 1}{2} - \frac{36 + x ^{2}}{6} : - 3 ln \frac{1}{6} x^{2} + 36 + 1 + 3 ln \frac{1}{6} x^{2} + 36 - 1 + 36 + x^{2}

= - 3 ln \frac{1}{6} x^{2} + 36 + 1 + 3 ln \frac{1}{6} x^{2} + 36 - 1 + 36 + x^{2}

解答补常数

= - 3 ln \frac{1}{6} x^{2} + 36 + 1 + 3 ln \frac{1}{6} x^{2} + 36 - 1 + 36 + x^{2} + C

作图

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