Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Integrals

Multiple Choice Questions

511.

If I₁ = $\int_{0}^{\frac{π}{2}} xsin (x) dx$ , I₂ = $\int_{0}^{\frac{π}{2}} xcos (x) dx$ , then whi ch one f the followin is true ?

I₁ = I₂
I₁ + I₂ = 0
$I_{1} = \frac{π}{2} . I_{2}$
$I_{1} + I_{2} = \frac{π}{2}$

512.

The value of $\int_{- 1}^{2} \frac{|x|}{x} dx$ , is

513.

$\int_{0}^{π} \frac{\cos^{4} (x)}{\cos^{4} (x) + \sin^{4} (x)} dx$ is equal to

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{8}$
$π$

514.

If f(x) = $f (π + e - x) and \int_{e}^{π} f (x) dx = \frac{2}{e + π}, then \int_{e}^{π} xf (x) dx$ is equal to

$π - e$
$\frac{π + e}{2}$
1
$\frac{π - e}{2}$

515.

If linear function f(x) and g(x) satisfy $\int [(3 x - 1) \cos (x) + (1 - 2 x) \sin (x)] dx$ = $f (x) \cos (x) + g (x) \sin (x) + C$ , then

f(x) = 3(x - 1)
f(x) = 3x - 5
g(x) = 3(x - 1)
g(x) = 3 + x

516.

The value of the integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} \log (\sec (θ) - \tan (θ)) dθ$ is

0
$\frac{π}{4}$
$π$
$\frac{π}{2}$

517.

$\int \frac{\sin (2 x)}{\sin^{2} (x) + 2 \cos^{2} (x)} dx$ is equal to

$- \log (1 + \sin^{2} (x)) + C$
$\log (1 + \cos^{2} (x)) + C$
$- \log (1 + \cos^{2} (x)) + C$
$\log (1 + \tan^{2} (x)) + C$

518.

$\int \frac{1}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}} dx$ is equal to

$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{2 x} + C$
$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{x} + C$
$\frac{- {(1 + x^{4})}^{\frac{3}{4}}}{x} + C$
$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{x^{2}} + C$

519.
$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$ is equal to

$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$

$\frac{π}{4} \log (2)$

$\log (2)$

$\frac{π}{2} \log (2)$

$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$

$I = \int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx \Rightarrow I = \int_{0}^{\frac{π}{4}} \log (1 + \tan (x)) dx . . . (i) = \int_{0}^{\frac{π}{4}} \log [1 + \tan (\frac{π}{4} - x)] dx = \int_{0}^{\frac{π}{4}} \log (1 + \frac{\tan (\frac{π}{4}) - \tan (x)}{1 + \tan (\frac{π}{4}) \tan (x)}) dx = \int_{0}^{\frac{π}{4}} \log (1 + \frac{1 - \tan (x)}{1 + \tan (x)}) dx \Rightarrow I = \int_{0}^{\frac{π}{4}} \log (\frac{2}{1 + \tan (x)}) dx . . . (ii) On adding Eqs (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{4}} \log (2) dx = \log (2) (\frac{π}{4}) \Rightarrow I = \frac{π}{8} \log (2)$