What is the integral of (5x^3-18)^715x^2 ?

The integral of (5x^3-18)^715x^2 is 1/8 (5x^3-18)^8+C

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integral of (5x^3-18)^715x^2

\int (5 x^{3} - 18)^{7} 15 x^{2} d x

\frac{1}{8} (5 x^{3} - 18)^{8} + C

Solution steps

\int (5 x^{3} - 18)^{7} \cdot 15 x^{2} d x

Take the constant out:

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

= 15 \cdot \int (5 x^{3} - 18)^{7} x^{2} d x

Apply u-substitution

: \int \frac{u ^{7}}{15} d u

= 15 \cdot \int \frac{u ^{7}}{15} d u

Take the constant out:

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

= 15 \cdot \frac{1}{15} \cdot \int u^{7} d u

Apply the Power Rule

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

= 15 \cdot \frac{1}{15} \cdot \frac{u ^{8}}{8}

Substitute back

u = 5 x^{3} - 18

= 15 \cdot \frac{1}{15} \cdot \frac{( 5 x ^{3} - 18 ) ^{8}}{8}

Simplify

15 \cdot \frac{1}{15} \cdot \frac{( 5 x ^{3} - 18 ) ^{8}}{8} : \frac{1}{8} (5 x^{3} - 18)^{8}

= \frac{1}{8} (5 x^{3} - 18)^{8}

Add a constant to the solution

= \frac{1}{8} (5 x^{3} - 18)^{8} + C

\int \frac{4 e ^{x}}{e ^{2 x} + 10 e ^{x} + 25} d x

\frac{d y}{d x} + x^{2} y = 2 x^{2}

y^{^{'}} = (y + 2 t)^{2}

- 2 y - y^{2} d x + (3 + 2 x - x^{2}) d y = 0

y^{^{''}} - y^{^{'}} - 2 y = cos (x) - sin (2 x)

What is the integral of (5x^3-18)^715x^2 ?
The integral of (5x^3-18)^715x^2 is 1/8 (5x^3-18)^8+C