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Integrals

Multiple Choice Questions

791.

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

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

792.

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

793.

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

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

794.

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

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

795.

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

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

796.

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

797.

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

798.
$\int_{0}^{π} \frac{\cos^{4} (x)}{\cos^{4} (x) + \sin^{4} (x)} dx$ is equal to

$\frac{π}{4}$

$\frac{π}{2}$

$\frac{π}{8}$

$π$

$\frac{π}{2}$

$Let I = \int_{0}^{π} \frac{\cos^{4} (x)}{\cos^{4} (x) + \sin^{4} (x)} dx I = 2 \int_{0}^{\frac{π}{2}} \frac{\cos^{4} (x)}{\cos^{4} (x) + \sin^{4} (x)} dx . . . (i) [∵ \int_{0}^{2 a} f (x) dx = 2 \int_{0}^{a} f (x) dx, if f (2 a - x) = f (x)] = \int_{0}^{\frac{π}{2}} \frac{\cos^{4} (\frac{π}{2} - x)}{\cos^{4} (\frac{π}{2} - x) + \sin^{4} (\frac{π}{2} - x)} dx = 2 \int_{0}^{\frac{π}{2}} \frac{\sin^{4} (x)}{\cos^{4} (x) + \sin^{4} (x)} dx . . . (ii)$

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