First slide

Evaluation of definite integrals

Question

The value of the integral $\int_{0}^{n π + t} (| \cos x |$ + |sin x|) dx is

Moderate

A

$n$
B

$2 n + \sin t + \cos t$
C

$\cos t$
D

$\sin t - \cos t + 4 n + 1$

Solution

Since the period of $| \sin x | + | \cos x$ is $π / 2$
$\begin{array}{l} \int_{0}^{n π + t} (| \sin x | + | \cos x |) d x = 2 n \int_{0}^{π / 2} (| \sin x | + | \cos x |) d x + \\ \int_{0}^{t} (| \sin x | + | \cos x |) d x \\ = 2 n \int_{0}^{π / 2} (\sin x + \cos x) d x + \int_{0}^{t} (\sin x + \cos x) d x \\ = (2 n) (2) + \sin t - \cos t + 1 \\ = (4 n + 1) + \sin t - \cos t . \end{array}$

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