What is the integral of 2x(x^2-8)^{102} ?

The integral of 2x(x^2-8)^{102} is 1/103 (x^2-8)^{103}+C

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integral of 2x(x^2-8)^{102}

\int 2 x (x^{2} - 8)^{102} d x

\frac{1}{103} (x^{2} - 8)^{103} + C

Solution steps

\int 2 x (x^{2} - 8)^{102} d x

Take the constant out:

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

= 2 \cdot \int x (x^{2} - 8)^{102} d x

Apply u-substitution

: \int \frac{u ^{102}}{2} d u

= 2 \cdot \int \frac{u ^{102}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{102} d u

Apply the Power Rule

: \frac{u ^{103}}{103}

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{103}}{103}

Substitute back

u = x^{2} - 8

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 8 ) ^{103}}{103}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 8 ) ^{103}}{103} : \frac{1}{103} (x^{2} - 8)^{103}

= \frac{1}{103} (x^{2} - 8)^{103}

Add a constant to the solution

= \frac{1}{103} (x^{2} - 8)^{103} + C

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What is the integral of 2x(x^2-8)^{102} ?
The integral of 2x(x^2-8)^{102} is 1/103 (x^2-8)^{103}+C