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Q)

Evaluate the following problems using properties of integration: $\int\limits_{-1}^{1}\sin x\cos^{4} x dx$

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A)
Toolbox:
  • $\int \limits_{-a}^a f(x) dx=2 \int \limits_0^a f(x) dx $ if f is an even function
  • $\int \limits_{-a}^a f(x) dx=0 $ if f is an odd function
Given $\int \limits_{-1}^1 \sin x \cos ^4 x dx$
Step 1:
$f(x)= \sin x \cos ^4 x dx$ and $f(-x)=- \sin x \cos ^4 x $
Step 2:
$\therefore f(x)$ is an odd function
$\int \limits_{-1}^1 \sin x \cos ^4 x dx=0$
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