First slide

Evaluation of definite integrals

Question

$\int_{- \frac{3 π}{2}}^{- \frac{π}{2}} {(x + π)}^{3} + \cos^{2} (x + 3 π) d x =$

Moderate

A

$\frac{π}{2}$
B

$\frac{π}{4} - 1$
C

$\frac{π^{4}}{32}$
D

$\frac{π^{4}}{32} + \frac{π}{2}$

Solution

$x + π = t \to I = \int_{- \frac{π}{2}}^{\frac{π}{2}} (t^{3} + \cos^{2} t) d t$ $= 2 \int_{0}^{\frac{π}{2}} \cos^{2} t d t = 2 \cdot \frac{1}{2} \cdot \frac{π}{2} = \frac{π}{2}$

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