What is the integral of (x^3)/(sqrt((11)^2-x^2)) ?

The integral of (x^3)/(sqrt((11)^2-x^2)) is 1/3 (121-x^2)^{3/2}-121sqrt(121-x^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3)/(sqrt((11)^2-x^2))

\int \frac{x ^{3}}{( 11 ) ^{2} - x ^{2}} d x

\frac{1}{3} (121 - x^{2})^{\frac{3}{2}} - 121 121 - x^{2} + C

Solution steps

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

Simplify

= \int \frac{x ^{3}}{121 - x ^{2}} d x

Apply u-substitution

: \int u^{2} - 121 d u

= \int u^{2} - 121 d u

Apply the Sum Rule:

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

= \int u^{2} d u - \int 121 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 121 d u = 121 u

= \frac{u ^{3}}{3} - 121 u

Substitute back

u = 121 - x^{2}

= \frac{( 121 - x ^{2} ) ^{3}}{3} - 121 121 - x^{2}

Simplify

\frac{( 121 - x ^{2} ) ^{3}}{3} - 121 121 - x^{2} : \frac{1}{3} (121 - x^{2})^{\frac{3}{2}} - 121 121 - x^{2}

= \frac{1}{3} (121 - x^{2})^{\frac{3}{2}} - 121 121 - x^{2}

Add a constant to the solution

= \frac{1}{3} (121 - x^{2})^{\frac{3}{2}} - 121 121 - x^{2} + C

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

\int \frac{x}{3 - 2 x} d x

\int (5 x^{4} - 4 x^{3} + 3 x^{2}) d x

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

\int (7 - 2 x)^{10} d x

What is the integral of (x^3)/(sqrt((11)^2-x^2)) ?
The integral of (x^3)/(sqrt((11)^2-x^2)) is 1/3 (121-x^2)^{3/2}-121sqrt(121-x^2)+C