First slide

Evaluation of definite integrals

Question

If $f (x) = \int_{0}^{x} \sin^{8} t d t,$ then $f (x + π)$ equals

Moderate

A

$\frac{f (x)}{f (π)}$
B

$f (x) f (π)$
C

$f (x) + f (π)$
D

$f (x) - f (π)$

Solution

$\begin{array}{l} f (x + π) = \int_{0}^{x + π} \sin^{8} t d t \\ = \int_{0}^{π} \sin^{8} t d t + \int_{π}^{x + π} \sin^{8} t d t \\ = f (x) + \int_{0}^{x} \sin^{8} (π - u) d u = f (π) + f (x) . \end{array}$

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