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Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{\large\frac{\pi}{2}}\sin 2x \cos x dx$

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  • If $F(x)=\int \limits_a^x f(t)dt $ then $\int \limits_a^b f(x) dx=F(b)-F(a)$
Given $\int\limits_{0}^{\large\frac{\pi}{2}}\sin 2x \cos x dx$
Step 1:
$\int\limits_{0}^{\large\frac{\pi}{2}}\sin 2x \cos x dx=\int\limits_{0}^{\large\frac{\pi}{2}} 2 \sin x \cos ^2 x dx$
$\qquad=2\int\limits_{0}^{\large\frac{\pi}{2}} \cos^2 x \sin x dx$
Step 2:
$\qquad=-2 \large \frac {\cos ^3 x}{3} \bigg]_0^{\large\frac{\pi}{2}}$
$\qquad= \large\frac{2}{3}$
answered Aug 14, 2013 by meena.p

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