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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{3}}{2} d u

= 2 \cdot \int \frac{u ^{3}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{3} d u

Apply the Power Rule

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

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{4}}{4}

Substitute back

u = x^{2} + 5

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

Simplify

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

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

Add a constant to the solution

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

x \to - 1 lim (\frac{6 x + 5}{5 x - 6})

\frac{\partial}{\partial x} \frac{10 3 ( 5 x ) ^{2}}{3}

\frac{d y}{d x} - \frac{2 y}{x} = x^{- 1} y^{- 1}

\int (\frac{c x}{( x ^{2} + y ^{2} ) ^{\frac{3}{2}}}) d x

\frac{d}{d x} (\frac{1}{x ( ln ( x ) ) ^{p (x)}})

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