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

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{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

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