First slide

Evaluation of definite integrals

Question

The definite integral
$\int_{- 2}^{0} (x^{3} + 3 x^{2} + 3 x + 3 + (x + 1) \cos (x + 1)) d x$ equals

Moderate

A

– 4
B

0
C

4
D

6

Solution

$I = \int_{- 2}^{0} [x^{3} + 3 x^{2} + 3 x + 3 + (x + 1) \cos (x + 1)] d x$
$= \int_{- 2}^{0} [(x + 1)^{3} + 2 + (x + 1) \cos (x + 1)] d x$
Put $x + 1 = t$ therefore,
$\begin{array}{l} I = \int_{- 1}^{1} [t^{3} + 2 + t \cos t] d t \\ = 4 [∵ t^{3} + t \cos t is an odd function] . \end{array}$

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