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

The integral of (x^2)/((1-x^3)^7) is 1/(18(1-x^3)^6)+C

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

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

\frac{1}{18 ( 1 - x ^{3} ) ^{6}} + C

Solution steps

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

Apply u-substitution

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

= \int \frac{1}{3 ( 1 - u ) ^{7}} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{( 1 - u ) ^{7}} d u

Apply u-substitution

: \int - \frac{1}{v ^{7}} d v

= \frac{1}{3} \cdot \int - \frac{1}{v ^{7}} d v

Take the constant out:

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

= \frac{1}{3} (- \int \frac{1}{v ^{7}} d v)

Apply the Power Rule

: - \frac{1}{6 v ^{6}}

= \frac{1}{3} (- (- \frac{1}{6 v ^{6}}))

Substitute back

= \frac{1}{3} (- (- \frac{1}{6 ( 1 - x ^{3} ) ^{6}}))

Simplify

\frac{1}{3} (- (- \frac{1}{6 ( 1 - x ^{3} ) ^{6}})) : \frac{1}{18 ( 1 - x ^{3} ) ^{6}}

= \frac{1}{18 ( 1 - x ^{3} ) ^{6}}

Add a constant to the solution

= \frac{1}{18 ( 1 - x ^{3} ) ^{6}} + C

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What is the integral of (x^2)/((1-x^3)^7) ?
The integral of (x^2)/((1-x^3)^7) is 1/(18(1-x^3)^6)+C