First slide

Evaluation of definite integrals

Question

The value of $a$ in $(- π, 0)$ satisfying $\sin α + \int_{α}^{2 α} \cos 2 x d x = 0$ is

Moderate

A

$- π / 2$
B

$- π$
C

$- π / 3$
D

$- π / 4$

Solution

$\begin{array}{l} \sin α + \int_{a}^{2 a} \cos 2 x d x = 0 \\ \Rightarrow \sin α + \frac{1}{2} (\sin 4 α - \sin 2 α) = 0 \\ \Rightarrow \sin α [1 + \cos (3 α)] = 0 \end{array}$
As $- π < α < 0, \sin α \neq 0$ therefore
$\cos (3 α) = - 1 \Rightarrow α = - π / 3$ .

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