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Questions  >>  CBSE XII  >>  Math  >>  Integrals
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Q)

$\int \large\frac{(2\sin 2x-\cos x)}{(6-\cos 2x -4\sin x) }$$dx$

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A)
Toolbox:
  • $I=\int f(x) dx$
  • Put $x=g(t)$ then $dx=g'(t)$
  • Therefore $I=\int f(g(t))g'(t)dt$
  • $\int \large\frac{dx}{x^+a^2}=\frac{1}{a}$$\tan^{-1}\big(\Large\frac{x}{a}\big)$
  • $\int \large\frac{dx}{x}$$=\log \mid x\mid+c$
Step 1:
Let $I = \int \large\frac{ (2\sin2x - \cos x)}{(6-\cos 2x-4\sin x)}$$dx$
We know that $\sin 2x=2\sin x\cos x$
$\qquad=\int\large\frac {[4\sin x\cos x - \cos x] dx}{5+2\sin^2x-4\sin x}$
$\qquad=\int \large\frac{ [\cos x(4\sin x-1)]dx}{[2\sin^2x-4\sin x+5]}$
Step 2:
Put $\sin x = t$ then $\cos xdx = dt$
substituting this we get,
$\qquad=\int \large\frac{ [4t-1]dt}{2t^2-4t+5}$
$\qquad=\int \large\frac{[4t-4+3]dt}{2t^2-4t+5}$
$\qquad=\int \large\frac{[4t-4]dt}{[2t^2-4t+5]} +\frac{3dt}{[2t^2-4t+5]}$
Step 3:
Put $2t^2-4t+5= y$, then $[4t-4]dt = dy$
Therefore $\int \large\frac{ dy}{y }+ 3\int \large\frac{dt}{(t-1)^2+\bigg(\Large\frac{\sqrt 3}{\sqrt 2}\bigg)^2}$
Step 4:
On integrating we get,
$\log |y| + 3 \bigg[\large\frac{\Large\frac{\sqrt 3}{\sqrt 2\tan^{-1}(t-1)}}{\Large\frac{\sqrt 3}{\sqrt 2}}\bigg]+C$
Step 5:
Substituting for $y$ and $t$ we get
$\log|2\sin^2x-4\sin x+5| + \sqrt 3\sqrt 2\tan^{-1}\bigg[\Large\frac{(\sin x-1)}{\sqrt 3}\bigg]$$+C$
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