Evaluate the following integrals: x+7 3x^2+25x+28 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{x}+7}{3\text{x}^2+25\text{x}+28}\text{ dx}$

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Answer

$\text{I}=\int\frac{\text{x}+7}{3\text{x}^2+25\text{x}+28}\text{ dx}$

$=\int\frac{\text{x}+7}{3\text{x}^2+21\text{x}+4\text{x}+28}\text{ dx}$

$=\int\frac{\text{x}+7}{3\text{x}(\text{x}+7)+4(\text{x}+7)}\text{ dx}$

$=\int\frac{\text{x}+7}{(3\text{x}+4)(\text{x}+7)}\text{ dx}$

$=\int\frac{1}{(3\text{x}+4)}\text{ dx}$

$=\frac{1}{3}\ln|3\text{x}+4|+\text{C}$

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