integration without using $u$ substitution

Question

Evaluation of Integration $\displaystyle \int \frac{4}{x+9}dx$ without using $u$ substitution.

What i try

$$4\int\frac{(x+9)-x}{x+9}dx=4\int dx-4\int\frac{x}{x+9}dx$$

How do I solve it without using $u$ substitution . Help me please.

I did not understand how one can able to solve without using $u$ substitution.

Answer 1

Try using: $$ \int{\frac{a\cdot f'(x)}{f(x)}} = a\cdot \ln|f(x)|+C $$ Very useful to be aware of that.

Answer 2

Without using $u$-substitution, I can think of using that the integrand is a geometric progression in disguise. And further using the Taylor series of $\ln(1+x)$, we can get the antiderivative.

$$\int\frac{4/9}{1+x/9}\mathrm dx=\frac{4}{9}\int\sum_{i=0}^{\infty}\frac{(-1)^{i}}{9^{i}}x^{i}\mathrm dx=4\sum_{i=0}^{\infty}\frac{(-1)^{i}}{9^{i+1}}\frac{x^{i+1}}{i+1}=4\ln\left(1+\frac{x}{9}\right)+\text{const.}$$