What is the integral of 12x^2(2x^3+3)^7 ?

The integral of 12x^2(2x^3+3)^7 is 1/4 (3+2x^3)^8+C

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integral of 12x^2(2x^3+3)^7

\int 12 x^{2} (2 x^{3} + 3)^{7} d x

\frac{1}{4} (3 + 2 x^{3})^{8} + C

Solution steps

\int 12 x^{2} (2 x^{3} + 3)^{7} d x

Take the constant out:

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

= 12 \cdot \int x^{2} (2 x^{3} + 3)^{7} d x

Apply u-substitution

: \int \frac{u ^{7}}{6} d u

= 12 \cdot \int \frac{u ^{7}}{6} d u

Take the constant out:

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

= 12 \cdot \frac{1}{6} \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= 12 \cdot \frac{1}{6} \cdot \frac{u ^{8}}{8}

Substitute back

u = 3 + 2 x^{3}

= 12 \cdot \frac{1}{6} \cdot \frac{( 3 + 2 x ^{3} ) ^{8}}{8}

Simplify

12 \cdot \frac{1}{6} \cdot \frac{( 3 + 2 x ^{3} ) ^{8}}{8} : \frac{1}{4} (3 + 2 x^{3})^{8}

= \frac{1}{4} (3 + 2 x^{3})^{8}

Add a constant to the solution

= \frac{1}{4} (3 + 2 x^{3})^{8} + C

\int \frac{x + 5}{x} d x

\int \frac{26 x ^{3} - 52 x ^{2} + 8}{x ^{2} - 2 x} d x

\int cot (16 x) d x

x \to 0 - lim (\frac{5}{x ^{3}})

x^{18}

What is the integral of 12x^2(2x^3+3)^7 ?
The integral of 12x^2(2x^3+3)^7 is 1/4 (3+2x^3)^8+C