What is the integral of (x^4)/(sqrt(x^5+1)) ?

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

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integral of (x^4)/(sqrt(x^5+1))

\int \frac{x ^{4}}{x ^{5} + 1} d x

\frac{2}{5} x^{5} + 1 + C

Solution steps

\int \frac{x ^{4}}{x ^{5} + 1} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{5} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{5} + 1

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

Simplify

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

= \frac{2}{5} x^{5} + 1

Add a constant to the solution

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

\int 16 d x

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\int (\frac{1}{3 + 2 x ^{2}}) d x

t \to 0 + lim (\frac{sin ( t )}{t})

\frac{d}{d x} (x^{3} (4 - x^{2}))

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