What is the integral of x^3sqrt(5x^4+20) ?

The integral of x^3sqrt(5x^4+20) is 1/30 (5x^4+20)^{3/2}+C

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integral of x^3sqrt(5x^4+20)

\int x^{3} 5 x^{4} + 20 d x

\frac{1}{30} (5 x^{4} + 20)^{\frac{3}{2}} + C

Solution steps

\int x^{3} 5 x^{4} + 20 d x

Apply u-substitution

: \int \frac{u ^{2}}{10} d u

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

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{1}{10} \cdot \frac{u ^{3}}{3}

Substitute back

u = 5 x^{4} + 20

= \frac{1}{10} \cdot \frac{( 5 x ^{4} + 20 ) ^{3}}{3}

Simplify

\frac{1}{10} \cdot \frac{( 5 x ^{4} + 20 ) ^{3}}{3} : \frac{1}{30} (5 x^{4} + 20)^{\frac{3}{2}}

= \frac{1}{30} (5 x^{4} + 20)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{30} (5 x^{4} + 20)^{\frac{3}{2}} + C

n = 0 \sum \infty (\frac{1}{17})^{n}

f (x) = \frac{4 x + 5}{x - 1}, at x = 2

x \to 0 lim (\frac{sin ( 5 x )}{6 x})

f (x) = 3 sec (5 x)

\frac{\partial}{\partial x} (\frac{17}{2} x^{4} y)

What is the integral of x^3sqrt(5x^4+20) ?
The integral of x^3sqrt(5x^4+20) is 1/30 (5x^4+20)^{3/2}+C