What is the integral of (x+7)/(sqrt(4-(x-2)^2)) ?

The integral of (x+7)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+9arcsin(1/2 x-1)+C

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integral of (x+7)/(sqrt(4-(x-2)^2))

\int \frac{x + 7}{4 - ( x - 2 ) ^{2}} d x

- - x^{2} + 4 x + 9 arcsin (\frac{1}{2} x - 1) + C

Solution steps

\int \frac{x + 7}{4 - ( x - 2 ) ^{2}} d x

Apply u-substitution

= \int \frac{u + 9}{4 - u ^{2}} d u

Apply Trigonometric Substitution

= \int 2 sin (v) + 9 d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 sin (v) d v + \int 9 d v

\int 2 sin (v) d v = - 2 cos (v)

\int 9 d v = 9 v

= - 2 cos (v) + 9 v

Substitute back

= - 2 cos (arcsin (\frac{1}{2} (x - 2))) + 9 arcsin (\frac{1}{2} (x - 2))

Simplify

- 2 cos (arcsin (\frac{1}{2} (x - 2))) + 9 arcsin (\frac{1}{2} (x - 2)) : - - x^{2} + 4 x + 9 arcsin (\frac{1}{2} x - 1)

= - - x^{2} + 4 x + 9 arcsin (\frac{1}{2} x - 1)

Add a constant to the solution

= - - x^{2} + 4 x + 9 arcsin (\frac{1}{2} x - 1) + C

What is the integral of (x+7)/(sqrt(4-(x-2)^2)) ?
The integral of (x+7)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+9arcsin(1/2 x-1)+C