What is the integral of 8x(4x^2+5)^3 ?

The integral of 8x(4x^2+5)^3 is 1/4 (4x^2+5)^4+C

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integral of 8x(4x^2+5)^3

\int 8 x (4 x^{2} + 5)^{3} d x

\frac{1}{4} (4 x^{2} + 5)^{4} + C

Solution steps

\int 8 x (4 x^{2} + 5)^{3} d x

Take the constant out:

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

= 8 \cdot \int x (4 x^{2} + 5)^{3} d x

Apply u-substitution

: \int \frac{u ^{3}}{8} d u

= 8 \cdot \int \frac{u ^{3}}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 8 \cdot \frac{1}{8} \cdot \frac{u ^{4}}{4}

Substitute back

u = 4 x^{2} + 5

= 8 \cdot \frac{1}{8} \cdot \frac{( 4 x ^{2} + 5 ) ^{4}}{4}

Simplify

8 \cdot \frac{1}{8} \cdot \frac{( 4 x ^{2} + 5 ) ^{4}}{4} : \frac{1}{4} (4 x^{2} + 5)^{4}

= \frac{1}{4} (4 x^{2} + 5)^{4}

Add a constant to the solution

= \frac{1}{4} (4 x^{2} + 5)^{4} + C

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What is the integral of 8x(4x^2+5)^3 ?
The integral of 8x(4x^2+5)^3 is 1/4 (4x^2+5)^4+C