The anti-derivative of cos 5 x-cos 4 x-1-2 cos 3 x is

The correct option is C sin 2x/2 sin x+cThe given anti derivative =∫cos 5x+cos 4x/1 2 cos 3x dx =∫2 cos 9x/2 cos x/2/1 2 { 2 cos^2 ( 3x/2 ) 1 }dx =∫2 cos 9x ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

The anti-deri...

Question

The anti-derivative of $\frac{c o s 5 x + c o s 4 x}{1 - 2 c o s 3 x}$ is

\frac{s i n 2 x}{2} + c o s x + c

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- \frac{s i n 2 x}{2} + s i n x + c

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- \frac{s i n 2 x}{2} - s i n x + c

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\frac{s i n 2 x}{2} - c o s x + c

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Solution

The correct option is C $- \frac{s i n 2 x}{2} - s i n x + c$
The given anti-derivative $= \int \frac{c o s 5 x + c o s 4 x}{1 - 2 c o s 3 x} d x$
$= \int \frac{2 c o s \frac{9 x}{2} c o s \frac{x}{2}}{1 - 2 {2 c o s^{2} (\frac{3 x}{2}) - 1}} d x = \int \frac{2 c o s \frac{9 x}{2} c o s \frac{x}{2}}{3 - 4 c o s^{2} \frac{3 x}{2}} d x = \int \frac{2 c o s \frac{9 x}{2} c o s \frac{x}{2} c o s \frac{3 x}{2}}{3 c o s \frac{3 x}{2} - 4 c o s^{3} \frac{3 x}{2}} d x$
[Multiplying and dividing by $c o s \frac{3 x}{2}$ ]
$= \int \frac{2 c o s \frac{9 x}{2} c o s \frac{3 x}{2} c o s \frac{x}{2}}{- c o s \frac{9 x}{2}} d x = [c o s 3 x = 4 c o s^{3} x - 3 c o s x] = - \int 2 c o s \frac{3 x}{2} c o s \frac{x}{2} = - \int (c o s 2 x + c o s x) d x = - \frac{s i n 2 x}{2} - s i n x + c$

Suggest Corrections

The anti-derivative of cos 5x+cos 4x1−2 cos 3x is

The anti-derivative of $\frac{c o s 5 x + c o s 4 x}{1 - 2 c o s 3 x}$ is