What is the integral of (7x)/((4x^2-1)^3) ?

The integral of (7x)/((4x^2-1)^3) is -7/(16(4x^2-1)^2)+C

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integral of (7x)/((4x^2-1)^3)

\int \frac{7 x}{( 4 x ^{2} - 1 ) ^{3}} d x

- \frac{7}{16 ( 4 x ^{2} - 1 ) ^{2}} + C

Solution steps

\int \frac{7 x}{( 4 x ^{2} - 1 ) ^{3}} d x

Take the constant out:

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

= 7 \cdot \int \frac{x}{( 4 x ^{2} - 1 ) ^{3}} d x

Apply u-substitution

: \int \frac{1}{8 u ^{3}} d u

= 7 \cdot \int \frac{1}{8 u ^{3}} d u

Take the constant out:

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

= 7 \cdot \frac{1}{8} \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

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

= 7 \cdot \frac{1}{8} (- \frac{1}{2 u ^{2}})

Substitute back

u = 4 x^{2} - 1

= 7 \cdot \frac{1}{8} (- \frac{1}{2 ( 4 x ^{2} - 1 ) ^{2}})

Simplify

7 \cdot \frac{1}{8} (- \frac{1}{2 ( 4 x ^{2} - 1 ) ^{2}}) : - \frac{7}{16 ( 4 x ^{2} - 1 ) ^{2}}

= - \frac{7}{16 ( 4 x ^{2} - 1 ) ^{2}}

Add a constant to the solution

= - \frac{7}{16 ( 4 x ^{2} - 1 ) ^{2}} + C

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What is the integral of (7x)/((4x^2-1)^3) ?
The integral of (7x)/((4x^2-1)^3) is -7/(16(4x^2-1)^2)+C