First slide

Introduction to integration

Question

If $\int \sqrt{2} \sqrt{1 + \sin x} dx = - 4 \cos (ax + b) + C,$ then the value of a, b are respectively

Moderate

A

$\frac{1}{2}, \frac{π}{4}$
B

$1, \frac{π}{2}$
C

1,1
D

None of these

Solution

Let,
$\begin{aligned} I = \int \sqrt{2} \sqrt{1 + \sin x} dx = \sqrt{2} \int (\sin \frac{x}{2} + \cos \frac{x}{2}) dx \\ = 2 \int \sin (\frac{π}{4} + \frac{x}{2}) dx = - 4 \cos (\frac{x}{2} + \frac{π}{4}) + C \end{aligned}$
But, $I = - 4 \cos (ax + b) + C$
on comparing' we get
$a = \frac{1}{2}, b = \frac{π}{4}$

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