What is the integral of (x+3)/(sqrt(4-(x-2)^2)) ?

The integral of (x+3)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+5arcsin(1/2 x-1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+3)/(sqrt(4-(x-2)^2))

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

- - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1) + C

Solution steps

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

Apply u-substitution

= \int \frac{u + 5}{4 - u ^{2}} d u

Apply Trigonometric Substitution

= \int 2 sin (v) + 5 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 5 d v

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

\int 5 d v = 5 v

= - 2 cos (v) + 5 v

Substitute back

= - 2 cos (arcsin (\frac{1}{2} (x - 2))) + 5 arcsin (\frac{1}{2} (x - 2))

Simplify

- 2 cos (arcsin (\frac{1}{2} (x - 2))) + 5 arcsin (\frac{1}{2} (x - 2)) : - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1)

= - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1)

Add a constant to the solution

= - - x^{2} + 4 x + 5 arcsin (\frac{1}{2} x - 1) + C

What is the integral of (x+3)/(sqrt(4-(x-2)^2)) ?
The integral of (x+3)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+5arcsin(1/2 x-1)+C