State whether the given equation is true or false -px-qx2-px-qx-e5logex-e4 logex-e3logex-e2logex-sin 2x-cos 2x-sin 2x cos 2xdx -p-qx-logp-q-q-px-logq-p-2x-x3-3-tan x- x-

The correct option is A TrueLet nbsp;I=∫ nbsp; (p^ x +q^ x )^ 2 / p^ x .q^ x + e^ 5log_ e x e^ 4log_ e x / e^ 3log_ e x e^ 2log_ e x +sin ^ 2 x+cos ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 1

State whether...

Question

State whether the given equation is true or false
$\int \frac{(p^{x} + q^{x})^{2}}{p^{x} . q^{x}} + \frac{e^{5 l o g_{e} x} - e^{4 l o g_{e} x}}{e^{3 l o g_{e} x} - e^{2 l o g_{e} x}} + \frac{{sin}^{2} x + {cos}^{2} x}{{sin}^{2} x {cos}^{2} x} d x$
$= \frac{(p / q)^{x}}{l o g (p / q)} + \frac{(q / p)^{x}}{l o g (q / p)} + 2 x + \frac{x^{3}}{3} + tan x - cot x$ .

True

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False

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Solution

The correct option is A True
Let $I = \int \frac{(p^{x} + q^{x})^{2}}{p^{x} . q^{x}} + \frac{e^{5 l o g_{e} x} - e^{4 l o g_{e} x}}{e^{3 l o g_{e} x} - e^{2 l o g_{e} x}} + \frac{{sin}^{2} x + {cos}^{2} x}{{sin}^{2} x {cos}^{2} x} d x$
$\Rightarrow I = I_{1} + I_{2} + I_{3}$
Where
$I_{1} = \int \frac{(p^{x} + q^{x})^{2}}{p^{x} . q^{x}} d x = \int (\frac{p^{2 x}}{p^{x} q^{x}} + \frac{q^{2 x}}{p^{x} q^{x}} + \frac{2 p^{x} q^{x}}{p^{x} q^{x}}) d x$
$= \int (\frac{1}{q^{x} / p^{x}} + \frac{1}{p^{x} / q^{x}} + 2) d x = \frac{(p / q)^{x}}{l o g (p / q)} + \frac{(q / p)^{x}}{l o g (q / p)} + 2 x$
$I_{2} = \int \frac{e^{5 l o g_{e} x} - e^{4 l o g_{e} x}}{e^{3 l o g_{e} x} - e^{2 l o g_{e} x}} d x = \int \frac{x^{5} - x^{4}}{x^{3} - x^{2}} d x = \int x^{2} d x = \frac{x^{3}}{3}$
$I_{3} = \int \frac{{sin}^{2} x + {cos}^{2} x}{{sin}^{2} x {cos}^{2} x} d x = \int ({sec}^{2} x + {csc}^{2} x) d x = tan x - cot x$
Hence
$I = \frac{(p / q)^{x}}{l o g (p / q)} + \frac{(q / p)^{x}}{l o g (q / p)} + 2 x + \frac{x^{3}}{3} + tan x - cot x$

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State whether the given equation is true or false∫(px+qx)2px.qx+e5logex−e4logexe3logex−e2logex+sin2x+cos2xsin2xcos2xdx=(p/q)xlog(p/q)+(q/p)xlog(q/p)+2x+x33+tanx−cotx.