Home page for accesible maths 1 1 Further Integration 1.20 Two distinct linear factors 1.22 Example continued

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1.21 Example involving a quadratic

Example.

Find

\int\frac{4x+1}{x^{2}+4}\,dx

We split the integral into the separate terms $\int\frac{4x}{x^{2}+4}\,dx$ and $\int\frac{1}{x^{2}+4}\,dx$ . For the first integral we make the substitution $u={x^{2}+4.}$ Then the first integral becomes

\int\frac{4x\,dx}{x^{2}+4}={\int\frac{2\,du}{u}=\,}{2\log|u|+c=2\log(x^{2}+4)+% c.}

For the second integral we substitute $x={2\tan t,}$ obtaining

\int\frac{1\,dx}{x^{2}+4}={\int\frac{2\sec^{2}t\,dt}{4\sec^{2}t}=\frac{1}{2}t+% c^{\prime}=\frac{1}{2}\tan^{-1}\frac{x}{2}+c^{\prime}.}