What is the integral of (x+5)/(sqrt(4-(x-2)^2)) ?

The integral of (x+5)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+7arcsin(1/2 x-1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x+5)/(sqrt(4-(x-2)^2))

\int \frac{x + 5}{4 - ( x - 2 ) ^{2}} d x

- - x^{2} + 4 x + 7 arcsin (\frac{1}{2} x - 1) + C

Solution steps

\int \frac{x + 5}{4 - ( x - 2 ) ^{2}} d x

Apply u-substitution

: \int \frac{u + 7}{4 - u ^{2}} d u

= \int \frac{u + 7}{4 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 2 sin (v) + 7 d v

= \int 2 sin (v) + 7 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 7 d v

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

\int 7 d v = 7 v

= - 2 cos (v) + 7 v

Substitute back

= - 2 cos (arcsin (\frac{1}{2} (x - 2))) + 7 arcsin (\frac{1}{2} (x - 2))

Simplify

- 2 cos (arcsin (\frac{1}{2} (x - 2))) + 7 arcsin (\frac{1}{2} (x - 2)) : - - x^{2} + 4 x + 7 arcsin (\frac{1}{2} x - 1)

= - - x^{2} + 4 x + 7 arcsin (\frac{1}{2} x - 1)

Add a constant to the solution

= - - x^{2} + 4 x + 7 arcsin (\frac{1}{2} x - 1) + C

\int sin^{2} (9 x) cos^{2} (9 x) d x

\int \frac{4}{15} x^{\frac{5}{2}} + 6 cos (x) + C x + D d x

\frac{d}{d x} (x^{7} ln (3 x))

\int_{1}^{2} 5 r^{2} ln (r) d r

\frac{d}{d x} ((- 3 cos (2 x) - sin (3 x))^{2})

What is the integral of (x+5)/(sqrt(4-(x-2)^2)) ?
The integral of (x+5)/(sqrt(4-(x-2)^2)) is -sqrt(-x^2+4x)+7arcsin(1/2 x-1)+C