What is the integral of x(1-x^2)^5 ?

The integral of x(1-x^2)^5 is -1/12 (1-x^2)^6+C

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integral of x(1-x^2)^5

\int x (1 - x^{2})^{5} d x

- \frac{1}{12} (1 - x^{2})^{6} + C

Solution steps

\int x (1 - x^{2})^{5} d x

Apply u-substitution

: \int - \frac{u ^{5}}{2} d u

= \int - \frac{u ^{5}}{2} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{6}}{6}

= - \frac{1}{2} \cdot \frac{u ^{6}}{6}

Substitute back

u = 1 - x^{2}

= - \frac{1}{2} \cdot \frac{( 1 - x ^{2} ) ^{6}}{6}

Simplify

- \frac{1}{2} \cdot \frac{( 1 - x ^{2} ) ^{6}}{6} : - \frac{1}{12} (1 - x^{2})^{6}

= - \frac{1}{12} (1 - x^{2})^{6}

Add a constant to the solution

= - \frac{1}{12} (1 - x^{2})^{6} + C

x = - 5, x = 3, y = 2 x^{2} + 12, y = 0

\int \frac{1}{( x - 2 ) ( x - 5 )} d x

\int 6 x (4 x^{2} - 9)^{2} d x

f (x) = 3 x - 0.6 x ln (x)

x \to 2 lim (x^{2} - 8 x - 2)

What is the integral of x(1-x^2)^5 ?
The integral of x(1-x^2)^5 is -1/12 (1-x^2)^6+C