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Integrals

Multiple Choice Questions

331.

$\int_{0}^{1} \frac{x^{2}}{1 + x^{2}} dx$ is equal to

$\frac{π}{4} - 1$
$1 - \frac{π}{2}$
$\frac{π}{2} - 1$
$1 - \frac{π}{4}$

332.

Minimize : z = 3x + y, subject to 2x + 3y $\leq$ 6, x + y $\geq$ 1, $x \geq 0, y \geq 0$

x = 1, y = 1
x = 0, y = 1
x = 1, y = 0
x = - 1, y = - 1

333.

$\int \frac{dx}{x (x^{7} + 1)}$ is equal to

$\log (\frac{x^{7}}{x^{7} + 1})$
$\frac{1}{7} \log (\frac{x^{7}}{x^{7} + 1}) + c$
$\log (\frac{x^{7} + 1}{x^{7}}) + c$
$\frac{1}{7} \log (\frac{x^{7} + 1}{x^{7}})$

334.

Using Trapezoidal rule and following table $\int_{0}^{8} f (x) dx$ is equal to

x 0 0 4 6 8

f(x) 2 5 10 17 26

184
92
46
- 36

335.

$\int \frac{dx}{x + \sqrt{x}}$ is equal to

$\frac{1}{2} \log (1 + \sqrt{x}) + c$
$2 \log (1 + \sqrt{x}) + c$
$\frac{1}{4} \log (1 + \sqrt{x}) + c$
$3 \log (1 + \sqrt{x}) + c$

336.

The value of $\int \frac{x^{2}}{1 + x^{6}} dx$ is

x³ + c
$\frac{1}{3} \tan^{- 1} (x^{3}) + c$
log(1 + x³) + c
None of these

337.

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

$\tan (\frac{x}{2}) + c$
$- \cot (\frac{x}{2}) + c$
$2 \tan (\frac{x}{2}) + c$
$- 2 \cot (\frac{x}{2}) + c$

338.

$\int \frac{\cos (x)}{\sqrt{1 + \sin (x)}} dx$ is equal to

$\sin (\frac{x}{2}) - \cos (\frac{x}{2})$
$\sin (\frac{x}{2}) + \cos (\frac{x}{2})$
$2 [\sin (\frac{x}{2}) - \cos (\frac{x}{2})] + c$
$2 [\sin (\frac{x}{2}) + \cos (\frac{x}{2})] + c$

339.

$\int_{1}^{3} \frac{\cos (\log (x))}{x} dx$ is equal to

1
$\cos (\log (3))$
$\sin (\log (3))$
$\frac{π}{4}$

340.
$\int_{0}^{\frac{π}{2}} \frac{\sin (x) + \cos (x)}{\sqrt{1 + \sin (2 x)}} dx$ is equal to

$- \frac{π}{2}$

$\frac{π}{4}$

$\frac{π}{2}$

$π$

$\frac{π}{2}$

$Let I = \int_{}^{\frac{π}{2}} \frac{\sin (x) + \cos (x)}{\sqrt{1 + \sin (2 x)}} dx = \int_{0}^{\frac{π}{2}} \frac{\sin (x) + \cos (x)}{\sqrt{\sin^{2} (x) + \cos^{2} (x) + 2 \sin (x) \cos (x)}} dx = \int_{0}^{\frac{π}{2}} \frac{(\sin (x) + \cos (x))}{\sqrt{{(\sin (x) + \cos (x))}^{2}}} dx = \int_{0}^{\frac{π}{2}} \frac{(\sin (x) + \cos (x))}{(\sin (x) + \cos (x))} dx = \int_{0}^{\frac{π}{2}} 1 dx = {[x]}_{0}^{\frac{π}{2}} = \frac{π}{2}$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

340. ∫0π2sinx + cosx1 + sin2xdx is equal to - π2 π4 π2 π

340.
$\int_{0}^{\frac{π}{2}} \frac{\sin (x) + \cos (x)}{\sqrt{1 + \sin (2 x)}} dx$ is equal to

$- \frac{π}{2}$

$\frac{π}{4}$

$\frac{π}{2}$

$π$