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受欢迎的微积分 >

integral of sqrt(5+x^2)

解答

\int 5 + x^{2} d x

解答

5 (\frac{1}{10} x 5 + x^{2} + \frac{1}{2} ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})) + C

求解步骤

\int 5 + x^{2} d x

使用三角换元法

= \int 5 sec^{3} (u) d u

提出常数:

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

= 5 \cdot \int sec^{3} (u) d u

使用降次积分法

= 5 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 5 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

u = arctan (\frac{1}{5} x)

代回

= 5 \frac{sec ^{2} ( arctan ( \frac{1}{5} x ) ) sin ( arctan ( \frac{1}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))

化简

5 \frac{sec ^{2} ( arctan ( \frac{1}{5} x ) ) sin ( arctan ( \frac{1}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) : 5 (\frac{1}{10} x 5 + x^{2} + \frac{1}{2} ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}))

= 5 (\frac{1}{10} x 5 + x^{2} + \frac{1}{2} ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}))

解答补常数

= 5 (\frac{1}{10} x 5 + x^{2} + \frac{1}{2} ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})) + C

作图

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