What is the integral of x(5x^2+9)^8 ?

The integral of x(5x^2+9)^8 is 1/90 (5x^2+9)^9+C

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integral of x(5x^2+9)^8

\int x (5 x^{2} + 9)^{8} d x

\frac{1}{90} (5 x^{2} + 9)^{9} + C

Solution steps

\int x (5 x^{2} + 9)^{8} d x

Apply u-substitution

: \int \frac{u ^{8}}{10} d u

= \int \frac{u ^{8}}{10} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int u^{8} d u

Apply the Power Rule

: \frac{u ^{9}}{9}

= \frac{1}{10} \cdot \frac{u ^{9}}{9}

Substitute back

u = 5 x^{2} + 9

= \frac{1}{10} \cdot \frac{( 5 x ^{2} + 9 ) ^{9}}{9}

Simplify

\frac{1}{10} \cdot \frac{( 5 x ^{2} + 9 ) ^{9}}{9} : \frac{1}{90} (5 x^{2} + 9)^{9}

= \frac{1}{90} (5 x^{2} + 9)^{9}

Add a constant to the solution

= \frac{1}{90} (5 x^{2} + 9)^{9} + C

y = e^{x^{5}}

\int sec (10 w) tan (10 w) d w

x \to 0 lim (\frac{x}{x ^{2} - 2 x})

\frac{d}{d x} (\frac{8 x}{5 - cot ( x )})

x \to 7 lim (2 x - ∣ x - 7 ∣)

What is the integral of x(5x^2+9)^8 ?
The integral of x(5x^2+9)^8 is 1/90 (5x^2+9)^9+C