What is the integral of (5x^4)/(sqrt(x^5+9)) ?

The integral of (5x^4)/(sqrt(x^5+9)) is 2sqrt(x^5+9)+C

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integral of (5x^4)/(sqrt(x^5+9))

integral

\frac{5 x ^{4}}{x ^{5} + 9}

2 x^{5} + 9 + C

Solution steps

\int \frac{5 x ^{4}}{x ^{5} + 9} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{4}}{x ^{5} + 9} d x

Apply u-substitution

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

= 5 \cdot \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

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

Apply the Power Rule

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

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

Substitute back

u = x^{5} + 9

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

Simplify

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

= 2 x^{5} + 9

Add a constant to the solution

= 2 x^{5} + 9 + C

x \cdot lo g_{10} (\frac{1}{x})

\frac{1}{( x ( ln ( x ) ) )}

u^{- 1}

\frac{( x ) ^{\frac{1}{2}}}{( a - x ) ^{\frac{1}{2}}}

4 y + 1

What is the integral of (5x^4)/(sqrt(x^5+9)) ?
The integral of (5x^4)/(sqrt(x^5+9)) is 2sqrt(x^5+9)+C