How do you find the integral of (4x)/(4x+7)dx?

Jun 17, 2015

I like rewriting: $\frac{4 x}{4 x + 7} = \frac{4 x + 7 - 7}{4 x + 7} = 1 - \frac{7}{4 x + 7}$

Explanation:

$\int \frac{4 x}{4 x + 7} = \int \left(1 - \frac{7}{4 x + 7}\right) \mathrm{dx}$

$= x - \frac{7}{4} \ln \left\mid 4 x + 7 \right\mid + C$

Note
To evaluate $\int \frac{7}{4 x + 7} \mathrm{dx}$ use substitution with $u = 4 x + 7$ so $\mathrm{dx} = \frac{1}{4} \mathrm{du}$ and we have $\frac{7}{4} \int \frac{1}{u} \mathrm{du}$

Second Note

This integral can also be evaluated by integration by parts, with $u = x$ and $\mathrm{dv} = \frac{4}{4 x + 7}$.

Parts gives us:

$x \ln \left\mid 4 x + 7 \right\mid - \int \ln \left(4 x + 7\right) \mathrm{dx}$

The integral here can be found by substitution if you know $\int \ln u \mathrm{du}$ and by substitution and integration by parts if you don't know $\int \ln u \mathrm{du}$.