What is the integral of x/(sqrt((x^2-25)^3)) ?

The integral of x/(sqrt((x^2-25)^3)) is -1/(sqrt(x^2-25))+C

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integral of x/(sqrt((x^2-25)^3))

\int \frac{x}{( x ^{2} - 25 ) ^{3}} d x

- \frac{1}{x ^{2} - 25} + C

Solution steps

\int \frac{x}{( x ^{2} - 25 ) ^{3}} d x

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{2}{u}

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

Substitute back

u = x^{2} - 25

= \frac{1}{2} (- \frac{2}{x ^{2} - 25})

Simplify

\frac{1}{2} (- \frac{2}{x ^{2} - 25}) : - \frac{1}{x ^{2} - 25}

= - \frac{1}{x ^{2} - 25}

Add a constant to the solution

= - \frac{1}{x ^{2} - 25} + C

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What is the integral of x/(sqrt((x^2-25)^3)) ?
The integral of x/(sqrt((x^2-25)^3)) is -1/(sqrt(x^2-25))+C