What is the integral of x(5-x^2)^{-3} ?

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

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integral of x(5-x^2)^{-3}

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

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

Solution steps

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

Simplify

x (5 - x^{2})^{- 3} : \frac{x}{( 5 - x ^{2} ) ^{3}}

= \int \frac{x}{( 5 - x ^{2} ) ^{3}} d x

Apply u-substitution

: \int - \frac{1}{2 u ^{3}} d u

= \int - \frac{1}{2 u ^{3}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{2} \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= - \frac{1}{2} (- \frac{1}{2 u ^{2}})

Substitute back

u = 5 - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\int 18 - 2 x^{2} d x

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\int - (x + 1) d x

\int 8^{2 x} d x

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