How to calculate $\int_{ }^{ }\frac{14x+2}{x^2+4}dx$

Question

This problem was given to me as part of a transcendental functions homework:

$$\int_{ }^{ }\frac{14x+2}{x^2+4}dx$$

I believe this problem requires substitution as well as sec-1 but i'm having difficulty piecing it together. Thanks!

Answer 1

$$\frac{14x+2}{x^2+4}=\frac{14x}{x^2+4}+\frac{2}{x^2+4}$$ And you can use now these: $$\int \frac{f'}{f}=\log(f)+C$$ And $$\int \frac{1}{x^2+1}\mathbb{d}x=\tan^{-1}(x)+C$$

Answer 2

We can split the integral up $\int\frac{14x+2}{x^{2}+4}=7\int\frac{2x}{x^{2}+4}+2\int\frac{1}{x^{2}+2^{2}}$ then we use standard results given in Botond's post above. The first integral is a log substitution and the second is a tan substitution. We get the solution $ln(x^{2}+4)+\frac{1}{2}tan^{-1}\frac{x}{2}+c$