What is the integral of (a^2)/(sqrt(4a-a^2)) ?

The integral of (a^2)/(sqrt(4a-a^2)) is 6arcsin(1/2 a-1)-sin(2arcsin(1/2 a-1))-4sqrt(-a^2+4a)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (a^2)/(sqrt(4a-a^2))

\int \frac{a ^{2}}{4 a - a ^{2}} d a

6 arcsin (\frac{1}{2} a - 1) - sin (2 arcsin (\frac{1}{2} a - 1)) - 4 - a^{2} + 4 a + C

Solution steps

\int \frac{a ^{2}}{4 a - a ^{2}} d a

Complete the square

4 a - a^{2} : - (a - 2)^{2} + 4

= \int \frac{a ^{2}}{- ( a - 2 ) ^{2} + 4} d a

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int (2 sin (v) + 2)^{2} d v

= \int (2 sin (v) + 2)^{2} d v

Expand

(2 sin (v) + 2)^{2} : 4 sin^{2} (v) + 8 sin (v) + 4

= \int 4 sin^{2} (v) + 8 sin (v) + 4 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 4 sin^{2} (v) d v + \int 8 sin (v) d v + \int 4 d v

\int 4 sin^{2} (v) d v = 2 v - sin (2 v)

\int 8 sin (v) d v = - 8 cos (v)

\int 4 d v = 4 v

= 2 v - sin (2 v) - 8 cos (v) + 4 v

Substitute back

= 2 arcsin (\frac{1}{2} (a - 2)) - sin (2 arcsin (\frac{1}{2} (a - 2))) - 8 cos (arcsin (\frac{1}{2} (a - 2))) + 4 arcsin (\frac{1}{2} (a - 2))

Simplify

2 arcsin (\frac{1}{2} (a - 2)) - sin (2 arcsin (\frac{1}{2} (a - 2))) - 8 cos (arcsin (\frac{1}{2} (a - 2))) + 4 arcsin (\frac{1}{2} (a - 2)) : 6 arcsin (\frac{1}{2} a - 1) - sin (2 arcsin (\frac{1}{2} a - 1)) - 4 - a^{2} + 4 a

= 6 arcsin (\frac{1}{2} a - 1) - sin (2 arcsin (\frac{1}{2} a - 1)) - 4 - a^{2} + 4 a

Add a constant to the solution

= 6 arcsin (\frac{1}{2} a - 1) - sin (2 arcsin (\frac{1}{2} a - 1)) - 4 - a^{2} + 4 a + C

\int 2 3 x^{4} d x

\int \frac{2}{4 + 2 x} d x

\int \frac{x}{( 4 x ^{2} + 9 ) ^{2}} d x

\int \frac{1}{25 - 9 x ^{2}} d x

\int \frac{8 x}{x ^{2} + 2} d x

What is the integral of (a^2)/(sqrt(4a-a^2)) ?
The integral of (a^2)/(sqrt(4a-a^2)) is 6arcsin(1/2 a-1)-sin(2arcsin(1/2 a-1))-4sqrt(-a^2+4a)+C