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受欢迎的微积分 >

integral of (x+6)/(sqrt(25-(x-5)^2))

解答

\int \frac{x + 6}{25 - ( x - 5 ) ^{2}} d x

解答

- - x^{2} + 10 x + 11 arcsin (\frac{1}{5} x - 1) + C

求解步骤

\int \frac{x + 6}{25 - ( x - 5 ) ^{2}} d x

使用换元积分法

= \int \frac{u + 11}{25 - u ^{2}} d u

使用三角换元法

= \int 5 sin (v) + 11 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

= \int 5 sin (v) d v + \int 11 d v

\int 5 sin (v) d v = - 5 cos (v)

\int 11 d v = 11 v

= - 5 cos (v) + 11 v

代回

= - 5 cos (arcsin (\frac{1}{5} (x - 5))) + 11 arcsin (\frac{1}{5} (x - 5))

化简

- 5 cos (arcsin (\frac{1}{5} (x - 5))) + 11 arcsin (\frac{1}{5} (x - 5)) : - - x^{2} + 10 x + 11 arcsin (\frac{1}{5} x - 1)

= - - x^{2} + 10 x + 11 arcsin (\frac{1}{5} x - 1)

解答补常数

= - - x^{2} + 10 x + 11 arcsin (\frac{1}{5} x - 1) + C

作图

流行的例子

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\int \frac{x ^{3} - x - 1}{( x + 1 ) ^{2}} d x

integral of 1/((x^2+1)(x-3)^2)

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integral of (x+(sin(sqrt(x)))/(sqrt(x)))

\int (x + \frac{sin ( x )}{x}) d x

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integral of 5x^2cos(7x)

\int 5 x^{2} cos (7 x) d x