What is the integral of (9x^4)/(4x^5+3) ?

The integral of (9x^4)/(4x^5+3) is 9/20 ln|4x^5+3|+C

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integral of (9x^4)/(4x^5+3)

\int \frac{9 x ^{4}}{4 x ^{5} + 3} d x

\frac{9}{20} ln 4 x^{5} + 3 + C

Solution steps

\int \frac{9 x ^{4}}{4 x ^{5} + 3} d x

Take the constant out:

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

= 9 \cdot \int \frac{x ^{4}}{4 x ^{5} + 3} d x

Apply u-substitution

= 9 \cdot \int \frac{1}{20 u} d u

Take the constant out:

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

= 9 \cdot \frac{1}{20} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 \cdot \frac{1}{20} ln ∣ u ∣

Substitute back

u = 4 x^{5} + 3

= 9 \cdot \frac{1}{20} ln 4 x^{5} + 3

Simplify

9 \cdot \frac{1}{20} ln 4 x^{5} + 3 : \frac{9}{20} ln 4 x^{5} + 3

= \frac{9}{20} ln 4 x^{5} + 3

Add a constant to the solution

= \frac{9}{20} ln 4 x^{5} + 3 + C

What is the integral of (9x^4)/(4x^5+3) ?
The integral of (9x^4)/(4x^5+3) is 9/20 ln|4x^5+3|+C