What is the integral of (5x)/((2-x^2)^3) ?

The integral of (5x)/((2-x^2)^3) is 5/(4(2-x^2)^2)+C

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integral of (5x)/((2-x^2)^3)

\int \frac{5 x}{( 2 - x ^{2} ) ^{3}} d x

\frac{5}{4 ( 2 - x ^{2} ) ^{2}} + C

Solution steps

\int \frac{5 x}{( 2 - x ^{2} ) ^{3}} 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} ) ^{3}} d x

Apply u-substitution

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

= 5 \cdot \int \frac{1}{2 ( 2 - u ) ^{3}} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{( 2 - u ) ^{3}} d u

Apply u-substitution

: \int - \frac{1}{v ^{3}} d v

= 5 \cdot \frac{1}{2} \cdot \int - \frac{1}{v ^{3}} d v

Take the constant out:

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

= 5 \cdot \frac{1}{2} (- \int \frac{1}{v ^{3}} d v)

Apply the Power Rule

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{5}{4 ( 2 - x ^{2} ) ^{2}} + C

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What is the integral of (5x)/((2-x^2)^3) ?
The integral of (5x)/((2-x^2)^3) is 5/(4(2-x^2)^2)+C