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Integrals

Multiple Choice Questions

501.

The value of $\int_{0}^{4} |x - 1| dx$ is

$\frac{5}{2}$
5
4
1

502.

If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (x) dx$ , where where n is apositive integer, then $I_{10} + I_{8}$ is

$\frac{1}{9}$
$\frac{1}{8}$
$\frac{1}{7}$
9

503.

If I_n = $\int e^{x} [\frac{\sin (x) + \cos (x)}{1 - \sin^{2} (x)}] dx$ is

$(e^{x} . \csc (x)) + C$
$(e^{x} . \cot (x)) + C$
$(e^{x} . \sec (x)) + C$
$(e^{x} . \tan (x)) + C$

504.

When x > 0, then $\int \cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) dx$ is

$2 [{xtan}^{- 1} (x) - \log (1 + x^{2})] + C$
$2 [{xtan}^{- 1} (x) + \log (1 + x^{2})] + C$
$2 {xtan}^{- 1} (x) + \log (1 + x^{2}) + C$
$2 {xtan}^{- 1} (x) - \log (1 + x^{2}) + C$

505.
If the area between y = mx² and x = my² (m > 0) is 1/4 sq units, then the value of m is

$\pm 3 \sqrt{2}$

$\pm \frac{2}{\sqrt{3}}$

$\sqrt{2}$

$\sqrt{3}$

$\pm \frac{2}{\sqrt{3}}$

$Given curves; y = {mx}^{2} and y^{2} m = x; m > 0 Intersection point of both curves x = m {({mx}^{2})}^{2} = m^{3} x^{4} \Rightarrow m^{3} x^{4} - x = 0 \Rightarrow x (m^{3} x^{3} - 1) = 0 \Rightarrow x (mx - 1) (m^{2} x^{2} + 1 + mx) = 0 \Rightarrow x = 0, x = 1 / m and y = 0, y = 1 / m We take only the points = (0, 0) and (1 / m, 1 / m) Now, the area of the curve = \int_{0}^{\frac{1}{m}} (\sqrt{\frac{x}{m}} - {mx}^{2}) dx Given, \frac{1}{4} = {[\frac{2}{3 \sqrt{m}} . x^{\frac{3}{2}} - m . \frac{x^{3}}{3}]}_{0}^{\frac{1}{m}} \Rightarrow \frac{1}{4} = [\frac{2}{3 \sqrt{m}} . \frac{1}{m^{\frac{3}{2}}} - \frac{m}{3} . \frac{1}{m^{3}}] \Rightarrow \frac{1}{4} = \{\frac{2}{3 m^{2}} - \frac{1}{3 m^{2}}\} \Rightarrow \frac{1}{4} = \frac{1}{3 m^{2}} \Rightarrow m^{2} = \frac{4}{3} ∴ m = \pm \frac{2}{\sqrt{3}}$

506.

$\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sin^{3} (x)}{\sin^{3} (x) + \cos^{3} (x)} dx$ is equal to

$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{12}$
$\frac{π}{6}$

507.

If [x] is the greatest integer function not greater than x, then $\int_{0}^{11} [x] dx$ is equal to

508.

If $n \in N$ and I_n = $\int {(\log (x))}^{n} dx$ , then I_n + nI_{n - 1} is equal to

$\frac{{(\log (x))}^{n + 1}}{n + 1}$
$x {(\log (x))}^{n} + C$
${(\log (x))}^{n - 1}$
$\frac{{(\log (x))}^{n}}{n}$

509.

$\int \frac{\cos^{n - 1} (x)}{\sin^{n + 1} (x)} dx (where, n \neq 0)$ is equal to

$\frac{\cot^{n} (x)}{n} + C$
$\frac{- \cot^{n - 1} (x)}{n - 1} + C$
$\frac{- \cot^{n} (x)}{n} + C$
$\frac{\cot^{n - 1} (x)}{n - 1} + C$

510.

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

$\frac{e^{x}}{(x + 1)} + C$
$\frac{e^{x}}{{(x + 1)}^{2}} + C$
$\frac{e^{x}}{{(x + 1)}^{3}} + C$
$\frac{{xe}^{x}}{(x + 1)} + C$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

505. If the area between y = mx2 and x = my2 (m > 0) is 1/4 sq units, then the value of m is ± 32 ± 23 2 3

505.
If the area between y = mx² and x = my² (m > 0) is 1/4 sq units, then the value of m is

$\pm 3 \sqrt{2}$

$\pm \frac{2}{\sqrt{3}}$

$\sqrt{2}$

$\sqrt{3}$