What is the integral of (4x)/((2x^2-7)^{10)} ?

The integral of (4x)/((2x^2-7)^{10)} is -1/(9(2x^2-7)^9)+C

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integral of (4x)/((2x^2-7)^{10)}

\int \frac{4 x}{( 2 x ^{2} - 7 ) ^{10}} d x

- \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}} + C

Solution steps

\int \frac{4 x}{( 2 x ^{2} - 7 ) ^{10}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{( 2 x ^{2} - 7 ) ^{10}} d x

Apply u-substitution

: \int \frac{1}{2 ( 2 u - 7 ) ^{10}} d u

= 4 \cdot \int \frac{1}{2 ( 2 u - 7 ) ^{10}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{( 2 u - 7 ) ^{10}} d u

Apply u-substitution

: \int \frac{1}{2 v ^{10}} d v

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{2 v ^{10}} d v

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{10}} d v

Apply the Power Rule

: - \frac{1}{9 v ^{9}}

= 4 \cdot \frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{9 v ^{9}})

Substitute back

= 4 \cdot \frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}})

Simplify

4 \cdot \frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}}) : - \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}}

= - \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}}

Add a constant to the solution

= - \frac{1}{9 ( 2 x ^{2} - 7 ) ^{9}} + C

3 t^{3}

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(9, 1), (- 1, 1)

\int - \frac{2}{x ^{3}} d x

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What is the integral of (4x)/((2x^2-7)^{10)} ?
The integral of (4x)/((2x^2-7)^{10)} is -1/(9(2x^2-7)^9)+C