What is the integral of (5x)/((2-x^2)^{13)} ?

The integral of (5x)/((2-x^2)^{13)} is 5/(24(2-x^2)^{12)}+C

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integral of (5x)/((2-x^2)^{13)}

\int \frac{5 x}{( 2 - x ^{2} ) ^{13}} d x

\frac{5}{24 ( 2 - x ^{2} ) ^{12}} + C

Solution steps

\int \frac{5 x}{( 2 - x ^{2} ) ^{13}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x}{( 2 - x ^{2} ) ^{13}} d x

Apply u-substitution

: \int - \frac{1}{2 u ^{13}} d u

= 5 \cdot \int - \frac{1}{2 u ^{13}} d u

Take the constant out:

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

= 5 (- \frac{1}{2} \cdot \int \frac{1}{u ^{13}} d u)

Apply the Power Rule

: - \frac{1}{12 u ^{12}}

= 5 (- \frac{1}{2} (- \frac{1}{12 u ^{12}}))

Substitute back

u = 2 - x^{2}

= 5 (- \frac{1}{2} (- \frac{1}{12 ( 2 - x ^{2} ) ^{12}}))

Simplify

5 (- \frac{1}{2} (- \frac{1}{12 ( 2 - x ^{2} ) ^{12}})) : \frac{5}{24 ( 2 - x ^{2} ) ^{12}}

= \frac{5}{24 ( 2 - x ^{2} ) ^{12}}

Add a constant to the solution

= \frac{5}{24 ( 2 - x ^{2} ) ^{12}} + C

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What is the integral of (5x)/((2-x^2)^{13)} ?
The integral of (5x)/((2-x^2)^{13)} is 5/(24(2-x^2)^{12)}+C