Resolved: "Evaluate the following integral: ∫sin⁡xsin⁡x+cos⁡xdx"

Michael Maggard

Answered question

2022-01-06

Evaluate the following integral:

\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} d x

Answer & Explanation

Fasaniu

Beginner2022-01-07Added 46 answers

I = \int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} d x = \int \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} d x

substitute

\tan x = t^{2}

and

d x = \frac{1}{1 + t^{4}} d t

I = \int \frac{t}{(1 + t) (1 + t^{4})} d t = \frac{1}{2} \int \frac{((1 + t^{4}) + (1 - t^{4})) t}{(1 + t) (1 + t^{4})} d t

= \frac{1}{2} \int \frac{t}{1 + t} d t + \frac{1}{2} \int \frac{(t - t^{2}) (1 + t^{2})}{1 + t^{4}} d t

= \frac{1}{2} \int \frac{t}{1 + t} d t + \frac{1}{2} \int \frac{(t - t^{2}) (1 + t^{2})}{1 + t^{4}} d t

= \frac{1}{2} \int \frac{(1 + t) - 1}{1 + t} d t + \frac{1}{2} \int \frac{t + t^{3} - (t^{2} - 1) - t^{4} - 1}{1 + t^{4}} d t

= - \frac{t}{2} + \frac{1}{2} \ln | t + 1 | + \frac{1}{4} \int \frac{2 t}{1 + t^{4}} + \frac{1}{2} \int \frac{t^{3}}{1 + t^{4}} d t - \frac{1}{2} \int \frac{t^{2} - 1}{1 + t^{4}} d t - \frac{1}{2} + C

All integrals are easy expect

This is helpful0Flag

usaho4w

Beginner2022-01-08Added 39 answers

We rationalise the denominator to get

I = \int \frac{\sin x - \sqrt{\cos x \sin x}}{\sin x - \cos x} d x

Writing everything in terms of

\cot x

, we get

I = \int \csc^{2} x (\frac{\sqrt{\cot x} - 1}{\cot^{3} x - \cot^{2} x + \cot x - 1}) d x

Now substituting

u = \cot x

and further

v = \sqrt{u}

gives us

I = - \int \frac{2 v}{v^{5} + v^{4} + v + 1} d v

Performing a partial fraction decomposition we have

I = \frac{2}{4 - 4 \sqrt{2}} \int \frac{v + \sqrt{2} - 1}{v^{2} + \sqrt{2} v + 1} d v + \frac{2}{4 + 4 \sqrt{2}}

\int \frac{v - \sqrt{2} - 1}{v^{2} - \sqrt{2} v + 1} d v + \int \frac{1}{1 + v} d v = I_{1} + I_{2} + I_{3}

Hope you can take it from here

star233

Skilled2022-01-11Added 403 answers

HINT multiply nominator and denominator by $\frac{1}{\sqrt{\sin (x)}}$ , then $t = \sqrt{\cot (x)}$ after all you'll have $\frac{2 t}{(t^{4} + 1) (t + 1)}$

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