If -cosec2x - 2010cos2010xdx-fxgx2010-C- then the number of solutions where equation fxgx- x in -0- 2- is-are-

The correct option is A 0 Given : ∫ (cosec^ 2 x 2010)dx cos ^ 2010 x #160; #160; Let x #160; cos ^ 2010 x #160; =t dt=cos ...

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Standard XII

Mathematics

Integration by Parts

If ∫cosec2x...

Question

If $\int \frac{{cosec}^{2} x - 2010}{{cos}^{2010 x}} d x = - \frac{f (x)}{(g (x))^{2010}} + C$ ; then the number of solutions where equation $\frac{f (x)}{g (x)} = {x}$ in $[0, 2 π]$ is/are:

0

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Solution

The correct option is A $0$
Given : $\int \frac{({cosec}^{2} x - 2010) d x}{{cos}^{2010} x}$
Let $\frac{cot x}{{cos}^{2010} x} = t d t = \frac{{cos}^{2010} x (- {cosec}^{2} x + 2010) d x}{({cos}^{2010} x)^{2}} - d t = \frac{({cosec}^{2} x - 2010) d x}{{cos}^{2010} x} \int - d t = - t + C = - \frac{cot x}{{cos}^{2010} x} f (x) = cot x g (x) = cos x \frac{f (x)}{g (x)} = cosec x = {x}$
Hence there is no solution.

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If ∫cosec2x−2010cos2010xdx=−f(x)(g(x))2010+C; then the number of solutions where equation f(x)g(x)={x} in [0,2π] is/are:

The correct option is A 0Given : ∫(cosec2x−2010)dxcos2010xLet cotxcos2010x=tdt=cos2010x(−cosec2x+2010)dx(cos2010x)2−dt=(cosec2x−2010)dxcos2010x∫−dt=−t+C=−cotxcos2010xf(x)=cotxg(x)=cosxf(x)g(x)=cosec x={x}Hence there is no solution.

If $\int \frac{{cosec}^{2} x - 2010}{{cos}^{2010 x}} d x = - \frac{f (x)}{(g (x))^{2010}} + C$ ; then the number of solutions where equation $\frac{f (x)}{g (x)} = {x}$ in $[0, 2 π]$ is/are: