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Integrals

Multiple Choice Questions

941.

$If f (t) = \int_{- t}^{t} \frac{e^{- |x|}}{2} dx, then \lim_{t \to \infty} f (t) = ?$

1
$\frac{1}{2}$
0
- 1

942.

$\int_{0}^{2 π} \sin^{6} (x) \cos^{5} (x) dx = ?$

$2 π$
$\frac{π}{2}$
0
$- π$

943.

If $\int e^{x} (\frac{1 - \sin (x)}{1 - \cos (x)}) dx = f (x) + constant$ , then f(x) is equal to

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

944.

If I_n = $\int x^{n} e^{cx} dx$ for n $\geq$ 1, then cI_n + n . I_n_{- 1} is equal to

xⁿe^cx
xⁿ
e^cx
xⁿ + e^cx

945.

If $\int e^{x} (1 + x) . \sec^{2} ({xe}^{x}) dx$ = f(x) + constant, then f(x) is equal to

$\cos ({xe}^{x})$
$\sin ({xe}^{x})$
$2 \tan^{- 1} (x)$
$\tan ({xe}^{x})$

946.
$\int_{0}^{1} x^{\frac{3}{2}} \sqrt{1 - x} dx$ is equal to

$\frac{π}{6}$

$\frac{π}{9}$

$\frac{π}{12}$

$\frac{π}{16}$

$\frac{π}{16}$

$Let I = \int_{0}^{1} x^{\frac{3}{2}} \sqrt{1 - x} dx Put x = \sin^{2} (θ) \Rightarrow dx = 2 \sin (θ) \cos (θ) dθ ∴ I = \int_{0}^{\frac{π}{2}} \sin^{3} (θ) \sqrt{1 - \sin^{2} (θ)} 2 \sin (θ) \cos (θ) dθ = 2 \int_{0}^{\frac{π}{2}} \sin^{4} (θ) \cos^{2} (θ) dθ = 2 [\frac{3 . 1 . 1}{6 . 4 . 2} . \frac{π}{2}] = \frac{π}{16}$