# Integral of (2y^3+3)/(9x+5)

## \int\frac{2y^3+3}{9x+5}dx

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$\frac{1}{3}\ln\left|5+9x\right|+\frac{2}{9}y^3\ln\left|5+9x\right|+C_0$

## Step by step solution

Problem

$\int\frac{2y^3+3}{9x+5}dx$
1

Split the fraction $\frac{2y^3+3}{5+9x}$ in two terms with same denominator

$\int\left(\frac{3}{5+9x}+\frac{2y^3}{5+9x}\right)dx$
2

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int\frac{3}{5+9x}dx+\int\frac{2y^3}{5+9x}dx$
3

Taking the constant out of the integral

$\int\frac{3}{5+9x}dx+y^3\int\frac{2}{5+9x}dx$
4

Apply the formula: $\int\frac{n}{b+x\cdot a}dx$$=\frac{n}{a}\ln\left|b+x\cdot a\right|, where a=9, b=5 and n=3 \frac{1}{3}\ln\left|5+9x\right|+y^3\int\frac{2}{5+9x}dx 5 Apply the formula: \int\frac{n}{b+x\cdot a}dx$$=\frac{n}{a}\ln\left|b+x\cdot a\right|$, where $a=9$, $b=5$ and $n=2$

$\frac{1}{3}\ln\left|5+9x\right|+\frac{2}{9}y^3\ln\left|5+9x\right|$
6

$\frac{1}{3}\ln\left|5+9x\right|+\frac{2}{9}y^3\ln\left|5+9x\right|+C_0$

$\frac{1}{3}\ln\left|5+9x\right|+\frac{2}{9}y^3\ln\left|5+9x\right|+C_0$

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### Main topic:

Integration by substitution

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