What is the integral of 24x(6x^2-1)^5 ?

The integral of 24x(6x^2-1)^5 is 1/3 (6x^2-1)^6+C

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integral of 24x(6x^2-1)^5

\int 24 x (6 x^{2} - 1)^{5} d x

\frac{1}{3} (6 x^{2} - 1)^{6} + C

Solution steps

\int 24 x (6 x^{2} - 1)^{5} d x

Take the constant out:

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

= 24 \cdot \int x (6 x^{2} - 1)^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{12} d u

= 24 \cdot \int \frac{u ^{5}}{12} d u

Take the constant out:

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

= 24 \cdot \frac{1}{12} \cdot \int u^{5} d u

Apply the Power Rule

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

= 24 \cdot \frac{1}{12} \cdot \frac{u ^{6}}{6}

Substitute back

u = 6 x^{2} - 1

= 24 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} - 1 ) ^{6}}{6}

Simplify

24 \cdot \frac{1}{12} \cdot \frac{( 6 x ^{2} - 1 ) ^{6}}{6} : \frac{1}{3} (6 x^{2} - 1)^{6}

= \frac{1}{3} (6 x^{2} - 1)^{6}

Add a constant to the solution

= \frac{1}{3} (6 x^{2} - 1)^{6} + C

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

What is the integral of 24x(6x^2-1)^5 ?
The integral of 24x(6x^2-1)^5 is 1/3 (6x^2-1)^6+C