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Integrals

Multiple Choice Questions

571.

$\int_{0}^{\frac{π}{2}} \frac{xsin (x) . \cos (x)}{\cos^{4} (x) + \sin^{4} (x)} dx$ is equal to

$\frac{π^{2}}{8}$
$\frac{π^{2}}{16}$
1
0

572.

Use Simpson's $\frac{1}{3}$ rule to find the value of $\int_{1}^{5} f (x) dx$ given,

x 1 2 3 4 5

y 10 50 70 80 100

140.88
256.66
160.26
None of these

573.

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

$xlog (1 + \cos (x)) + c$
$\frac{1}{x} \log (1 + \cos (x)) + c$
$xtan (\frac{x}{2}) + c$
$x^{2} \tan^{- 1} (\frac{x}{2}) + c$

574.

The value of $\int \frac{\cos (\sqrt{x})}{\sqrt{x}} dx$ will be

$2 \sin (\sqrt{x}) + c$
$2 \cos (\sqrt{x}) + c$
$2 \sin (x) + c$
$\sqrt{2} \sin (x) + c$

575.

The value of $\int_{2}^{3} \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx$ will be

$\frac{\sqrt{3}}{2}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$
$\frac{1}{\sqrt{3}}$

576.

The integral $\int_{1}^{10} x^{3} dx$ is approximately evaluated by Trapezoidal rule $\int_{1}^{10} x^{3} dx$ = $3 [\frac{1 + 10^{3}}{2} + α + 7^{3}]$ for n = 3, then the value of $α$ is

4³
4²
5³
None of these

577.

Let f(x) = $\int_{u (x)}^{v (x)} g (t) dt$ . Then, f'(x) is equal to

g[v(x)] - g[u(x)]
g'[v(x)] - g'[u(x)]
g[v(x)] v'(x) - g[u(x)] u'(x)
None of the above

578.
The value of $\int_{0}^{π} |\cos (x)| dx$ is

1

2

0

3

$\int_{0}^{π} |\cos (x)| dx = \int_{0}^{\frac{π}{2}} \cos (x) dx + \int_{\frac{π}{2}}^{π} (- \cos (x)) dx = {(\sin (x))}_{0}^{\frac{π}{2}} - {(\sin (x))}_{\frac{π}{2}}^{π} = 2 \sin (\frac{π}{2}) = 2$