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Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{\large\frac{\pi}{2}}\cos^{3}x dx$

$\begin{array}{1 1}\frac{2}{5} \\ \frac{5}{3} \\ \frac{2}{3} \\ \frac{7}{3} \end{array} $

1 Answer

Toolbox:
  • If $F(x)=\int \limits_a^x f(t)dt $ then $\int \limits_a^b f(x) dx=F(b)-F(a)$
Given $\int \limits_0^{\large\frac{\pi}{2}} \cos^3x dx$
Step 1:
$\int \limits_0^{\large\frac{\pi}{2}} \cos^3x dx=\int \limits_0^{\large\frac{\pi}{2}} \large\frac{\cos 3x+3 \cos x}{4}$$ dx$
Step 2:
$\qquad=\large\frac{\sin 3x}{12}+\frac{3 \sin x}{4} \bigg]_0^{\large\frac{\pi}{2}}$
$\qquad=-\large\frac{1}{12}+\frac{3}{4}$
$\qquad=\large\frac{8}{12}$
$\qquad=\large\frac{2}{3}$
answered Aug 14, 2013 by meena.p
 

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