What is the integral of 3(x^2-2)^22x ?

The integral of 3(x^2-2)^22x is (x^2-2)^3+C

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integral of 3(x^2-2)^22x

\int 3 (x^{2} - 2)^{2} 2 x d x

(x^{2} - 2)^{3} + C

Solution steps

\int 3 (x^{2} - 2)^{2} \cdot 2 x d x

Take the constant out:

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

= 3 \cdot 2 \cdot \int (x^{2} - 2)^{2} x d x

Simplify

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

Apply u-substitution

= 6 \cdot \int \frac{u ^{2}}{2} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int u^{2} d u

Apply the Power Rule

= 6 \cdot \frac{1}{2} \cdot \frac{u ^{3}}{3}

Substitute back

u = x^{2} - 2

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

Simplify

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

= (x^{2} - 2)^{3}

Add a constant to the solution

= (x^{2} - 2)^{3} + C

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What is the integral of 3(x^2-2)^22x ?
The integral of 3(x^2-2)^22x is (x^2-2)^3+C