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

The integral of x(2x^2+1)^5 is 1/24 (2x^2+1)^6+C

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

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

\frac{1}{24} (2 x^{2} + 1)^{6} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{5}}{4} d u

= \int \frac{u ^{5}}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u^{5} d u

Apply the Power Rule

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

= \frac{1}{4} \cdot \frac{u ^{6}}{6}

Substitute back

u = 2 x^{2} + 1

= \frac{1}{4} \cdot \frac{( 2 x ^{2} + 1 ) ^{6}}{6}

Simplify

\frac{1}{4} \cdot \frac{( 2 x ^{2} + 1 ) ^{6}}{6} : \frac{1}{24} (2 x^{2} + 1)^{6}

= \frac{1}{24} (2 x^{2} + 1)^{6}

Add a constant to the solution

= \frac{1}{24} (2 x^{2} + 1)^{6} + C

(- 3, 5), (2, - 1)

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

y^{^{'}} = \frac{( x ^{2} + 2 y ^{2} )}{x y}

a e^{- b^{2} x^{2}}

e^{2}

What is the integral of x(2x^2+1)^5 ?
The integral of x(2x^2+1)^5 is 1/24 (2x^2+1)^6+C