Practicing Success
Let $f$ be an integrable function defined on $[0, a]$. If $I_1=\int\limits_0^{\pi / 2} \cos \theta f\left(\sin \theta+\cos ^2 \theta\right) d \theta$ and, $I_2=\int\limits_0^{\pi / 2} \sin 2 \theta f\left(\sin \theta+\cos ^2 \theta\right) d \theta$, then |
$I_1=I_2$ $I_1=-I_2$ $I_1=2 I_2$ $I_1=-2 I_2$ |
$I_1=I_2$ |
We have, $I_1-I_2=\int\limits_0^{\pi / 2}(\cos \theta-\sin 2 \theta) f\left(\sin \theta+\cos ^2 \theta\right) d \theta$ $\Rightarrow I_1-I_2=\int\limits_0^1 f\left(\sin \theta+\cos ^2 \theta\right) d\left(\sin \theta+\cos ^2 \theta\right)$ $\Rightarrow I_1-I_2=\int\limits_1^1 f(t) d t=0$, where $t=\sin \theta+\cos ^2 \theta$ $\Rightarrow I_1=I_2$ |