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受欢迎的微积分 >

integral of x^2sqrt(9+x^2)

解答

\int x^{2} 9 + x^{2} d x

解答

81 (- \frac{1}{72} (x 9 + x^{2} + 9 ln (\frac{x + 9 + x ^{2}}{3})) + \frac{1}{324} x (x^{2} + 9)^{\frac{3}{2}}) + C

求解步骤

\int x^{2} 9 + x^{2} d x

使用三角换元法

= \int 81 sec^{3} (u) tan^{2} (u) d u

提出常数:

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

= 81 \cdot \int sec^{3} (u) tan^{2} (u) d u

使用三角恒等式改写

= 81 \cdot \int sec^{3} (u) (- 1 + sec^{2} (u)) d u

乘开

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= 81 \cdot \int - sec^{3} (u) + sec^{5} (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

= 81 (- \int sec^{3} (u) d u + \int sec^{5} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

\int sec^{5} (u) d u = \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

= 81 (- (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

u = arctan (\frac{1}{3} x)

代回

= 81 (- (\frac{1}{2} sec (arctan (\frac{1}{3} x)) tan (arctan (\frac{1}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{3} x)) + sec (arctan (\frac{1}{3} x))) + \frac{sec ^{4} ( arctan ( \frac{1}{3} x ) ) sin ( arctan ( \frac{1}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{1}{3} x)) tan (arctan (\frac{1}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{3} x)) + sec (arctan (\frac{1}{3} x))))

化简

81 (- (\frac{1}{2} sec (arctan (\frac{1}{3} x)) tan (arctan (\frac{1}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{3} x)) + sec (arctan (\frac{1}{3} x))) + \frac{sec ^{4} ( arctan ( \frac{1}{3} x ) ) sin ( arctan ( \frac{1}{3} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{1}{3} x)) tan (arctan (\frac{1}{3} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{3} x)) + sec (arctan (\frac{1}{3} x)))) : 81 (- \frac{1}{72} (x 9 + x^{2} + 9 ln (\frac{x + 9 + x ^{2}}{3})) + \frac{1}{324} x (x^{2} + 9)^{\frac{3}{2}})

= 81 (- \frac{1}{72} (x 9 + x^{2} + 9 ln (\frac{x + 9 + x ^{2}}{3})) + \frac{1}{324} x (x^{2} + 9)^{\frac{3}{2}})

解答补常数

= 81 (- \frac{1}{72} (x 9 + x^{2} + 9 ln (\frac{x + 9 + x ^{2}}{3})) + \frac{1}{324} x (x^{2} + 9)^{\frac{3}{2}}) + C

作图

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