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

The integral of (2x)/((2x^2-5)^2) is -1/(2(2x^2-5))+C

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integral of (2x)/((2x^2-5)^2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{4 u ^{2}} d u

= 2 \cdot \int \frac{1}{4 u ^{2}} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 2 \cdot \frac{1}{4} (- \frac{1}{u})

Substitute back

u = 2 x^{2} - 5

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

Simplify

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

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

Add a constant to the solution

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

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