JEE MAIN Indefinite Integration Previous Year Questions (PYQs) – Page 3 of 5
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Let $I_n=\int \tan^{n}x,dx,\ (n>1)$.
If $I_4+I_6=a\tan^{5}x+bx^{5}+C$, where $C$ is a constant of integration,
then the ordered pair $(a,b)$ is equal to :
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Solution
$\displaystyle \int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}},dx$ is equal to (where $c$ is a constant of integration):
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Solution
Let $f(x)=\displaystyle \int \frac{2x}{(x^{2}+1)(x^{2}+3)}\,dx$.
If $f(3)=\dfrac{1}{2}(\log_{e}5-\log_{e}6)$, then $f(4)$ is equal to:
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Solution
If
$\displaystyle \int \frac{\sin^{3/2}x+\cos^{3/2}x}{\sqrt{\sin^2 x,\cos^2 x},\sin(x-\theta)},dx
= A\sqrt{\cos\theta,\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta,\cot x}+C,$
where $C$ is the integration constant, then $AB$ is equal to:
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Solution
The integral
$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $
$\left( {0 < x < {\pi \over 2}} \right)$ is equal to :
(where C is a constant of integration)
$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $
$\left( {0 < x < {\pi \over 2}} \right)$ is equal to :
(where C is a constant of integration)
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Solution
If $\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$, where c is a constant of integration, thenthe ordered pair (a, b) is equal to :
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Solution
If $\displaystyle \int \frac{dx}{x^{3}(1+x^{6})^{2/3}}=x,f(x),(1+x^{6})^{1/3}+C$ where $C$ is a constant of integration, then the function $f(x)$ is equal to:
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Solution
Let
$ \displaystyle \int \frac{2 - \tan x}{3 + \tan x} , dx = \frac{1}{2} \left( \alpha x + \log_e \left| \beta \sin x + \gamma \cos x \right| \right) + C $,
where $C$ is the constant of integration.
Then $\alpha + \dfrac{\gamma}{\beta}$ is equal to:
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Solution
The integral $\displaystyle \int \sec^{2/3}x , \csc^{4/3}x , dx$ is equal to (Hence $C$ is a constant of integration)
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Solution
If
$\displaystyle f\left(\frac{3x-4}{3x+4}\right) = x + 2,; x \ne -\frac{4}{3}$
and
$\displaystyle \int f(x),dx = A\ln|1-x| + Bx + C,$
then the ordered pair $(A,B)$ is equal to
(where $C$ is a constant of integration):
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Solution
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