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

The integral of x(2x^2+3)^4 is 1/20 (2x^2+3)^5+C

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

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

\frac{1}{20} (2 x^{2} + 3)^{5} + C

Solution steps

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

Apply u-substitution

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

= \int \frac{u ^{4}}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{1}{4} \cdot \frac{u ^{5}}{5}

Substitute back

u = 2 x^{2} + 3

= \frac{1}{4} \cdot \frac{( 2 x ^{2} + 3 ) ^{5}}{5}

Simplify

\frac{1}{4} \cdot \frac{( 2 x ^{2} + 3 ) ^{5}}{5} : \frac{1}{20} (2 x^{2} + 3)^{5}

= \frac{1}{20} (2 x^{2} + 3)^{5}

Add a constant to the solution

= \frac{1}{20} (2 x^{2} + 3)^{5} + C

x \to 0 lim (\frac{sin ( x )}{- x})

\frac{d}{d y} (x^{\frac{y}{z}})

- sin (x^{2})

\int_{2}^{8} \frac{1}{2 x + 1} d x

\int_{0}^{1} 4 t d t

What is the integral of x(2x^2+3)^4 ?
The integral of x(2x^2+3)^4 is 1/20 (2x^2+3)^5+C