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tabita57i

Answered question

2021-02-02

Evaluate the integral

\int \frac{1}{1 + {\frac{x}{2}}^{2}} d x

pierretteA

Skilled2021-02-03Added 102 answers

The first step here will be a substitution to simplify that denominator.

u = 1 + \frac{x}{2}

∴ d u = \frac{x}{2} d x \to 2 d u = d x

Substituting these in, we get

\int \frac{1}{u^{2}} \times 2 d u = 2 \int u^{- 2} d u

Now we just use the power rule for integrals.

= 2 [\frac{u^{- 1}}{1}] + C = - \frac{2}{u} + c

And finally, we substitute back in for xx and simplify

= - \frac{2}{1 + \frac{x}{2} + C} = - \frac{4}{2 + x} + c

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