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

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

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

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

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

Solution steps

\int x (x^{2} + 1)^{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 = x^{2} + 1

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

Simplify

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

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

Add a constant to the solution

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

x = 3 y^{2}, x = 4 + 2 y^{2}

x \to - 4 lim (x^{2} - 3)

x^{2} y^{^{''}} + 3 x y^{^{'}} - 24 y = 0

\int \frac{4 x}{2 x + 3} d x

f (x) = 3 - 6 x

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