Caelan

2021-11-07

Evaluate the integral.
$\int \frac{\mathrm{sin}0}{1+{\mathrm{cos}}^{2}0}d0$

Step 1
To Determine:
Find or evaluate the integral
Given: we have an integral $\int \frac{\mathrm{sin}0}{1+{\mathrm{cos}}^{2}0}d0$
Explanation: we have $\int \frac{\mathrm{sin}0}{1+{\mathrm{cos}}^{2}0}d0$
let us consider $t=\mathrm{cos}0⇒dt=-\mathrm{sin}0d0$
so the integral becomes
$-\int \frac{1}{1+{t}^{2}}=-{\mathrm{tan}}^{-1}t+c$
Step 2
now putting the value of t then we have
$\int \frac{\mathrm{sin}0}{1+{\mathrm{cos}}^{2}0}d0=-{\mathrm{tan}}^{-1}\left(\mathrm{cos}0\right)+c$

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