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

841.

Intersection point of f₁(x) = $\int_{2}^{x} (2 t - 5) dt and f_{2} (x) = \int_{0}^{x} 2 tdt$ is

$(\frac{6}{5}, \frac{36}{25})$
$(\frac{2}{3}, \frac{4}{9})$
$(\frac{1}{3}, \frac{1}{9})$
$(\frac{1}{5}, \frac{1}{25})$

842.

The value of $\lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} e^{\frac{r}{n}}$ is

e
e - 1
1 - e
1 + e

843.

$\int_{0}^{2} \sqrt{\frac{2 + x}{2 - x}} dx$ is equal to

$π + 2$
$π + \frac{3}{2}$
$π + 1$
None of these

844.

$\int e^{x} \sin (e^{x}) dx$ is equal to

$- \cos (e^{x}) + c$
$\cos (e^{x}) + c$
$- \csc (e^{x}) + c$
None of these

845.

$\int e^{x} (\frac{1}{x} - \frac{1}{x^{2}}) dx$ is equal to

$\frac{- e^{x}}{x^{2}} + c$
$\frac{e^{x}}{x^{2}} + c$
$\frac{e^{x}}{x} + c$
$\frac{- e^{x}}{x} + c$

846.
$\int \sqrt{(\frac{1 - \sqrt{x}}{1 + \sqrt{x}})} dx$ is equal to

$\cos^{- 1} (\sqrt{x}) + \sqrt{1 - x} (\sqrt{x} - 2) + c$

$\cos^{- 1} (\sqrt{x}) - \sqrt{1 - x} (\sqrt{x} - 2) + c$

$\cos^{- 1} (\sqrt{x}) + \sqrt{1 - x} (\sqrt{x} + 2) + c$

None of the above

$\cos^{- 1} (\sqrt{x}) + \sqrt{1 - x} (\sqrt{x} - 2) + c$

$Let I = \int \sqrt{(\frac{1 - \sqrt{x}}{1 + \sqrt{x}})} dx Put x = \cos^{2} (θ) and dx = - 2 \sin (θ) \cos (θ) dθ ∴ I = \int \sqrt{(\frac{1 - \sqrt{\cos (θ)}}{1 + \sqrt{\cos (θ)}})} (- 2 \sin (θ) \cos (θ)) dθ = - 2 \int \sqrt{\frac{2 \sin^{2} (\frac{θ}{2})}{2 \cos^{2} (\frac{θ}{2})}} . \sin (\frac{θ}{2}) \cos (\frac{θ}{2}) \cos (θ) dθ = - 4 \int \sin^{2} (\frac{θ}{2}) \cos (θ) dθ = - 2 \int (1 - \cos (θ)) \cos (θ) dθ = - 2 \int (\cos (θ) - \frac{1 + \cos (2 θ)}{2}) dθ = θ + \frac{1}{2} \sin (2 θ) - 2 \sin (θ) = θ + \sin (θ) \cos (θ) - 2 \sin (θ) = \cos^{- 1} (\sqrt{x}) + \sqrt{1 - x} (\sqrt{x} - 2) + c$

847.

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

$\frac{1}{2} {[\log (x + \sqrt{1 + x^{2}})]}^{2} + c$
$\log {(x + \sqrt{1 + x^{2}})}^{2} + c$
$\log (x + \sqrt{1 + x^{2}}) + c$
None of the above

848.

$\int \frac{\sin^{8} (x) - \cos^{8} (x)}{1 - 2 \sin^{2} (x) \cos^{2} (x)} dx$ is equal to

$\sin (2 x) + c$
$- \frac{1}{2} \sin (2 x) + c$
$\frac{1}{2} \sin (2 x) + c$
$- \sin (2 x) + c$

849.

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

$\frac{1}{6} \log (1 - \cos (x)) + \frac{1}{2} \log (\log (1 + \cos (x))) - \frac{2}{3} \log (1 + 2 \cos (x)) + c$
$6 \log (1 - \cos (x)) + 2 \log (\log (1 + \cos (x))) - \frac{2}{3} \log (1 + 2 \cos (x)) + c$
$6 \log (1 - \cos (x)) + \frac{1}{2} \log (\log (1 + \cos (x))) + \frac{2}{3} \log (1 + 2 \cos (x)) + c$
None of the above