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Integrals

Multiple Choice Questions

31.

What is $\int_{- 2}^{2} x d x - \int_{- 2}^{2} [x] d x$

32.

If $\int_{- 2}^{5} f (x) d x = 4 and \int_{0}^{5} [1 + f (x)] d x = 7$

then what is $\int_{- 2}^{0} f (x) d x = ?$

33.

What is $\int_{0}^{4 π} |\cos (x)| dx$ = ?

34.

Let f(x) be a function such that $f' (\frac{1}{x}) + x^{3} f' (x) = 0 . What is \int_{- 1}^{1} f (x) d x = ?$

2f(1)
0
2f( - 1)
4f(1)

35.

What is $\int \frac{x^{4} - 1}{x^{2} \sqrt{x^{4} + x^{2} + 1}} dx = ?$

$\sqrt{\frac{x^{4} + x^{2} + 1}{4}} + c$
$\sqrt{x^{2} + 2 - \frac{1}{x^{2}}} + c$
$\sqrt{x^{2} + \frac{1}{x^{2}} + 1} + c$
$\sqrt{\frac{x^{4} - x^{2} + 1}{x}} + c$

36.

What is $\int e^{\sin (x)} \frac{{xcos}^{3} (x) - \sin (x)}{\cos^{2} (x)} dx = ?$

$(x + \sec (x)) e^{\sin (x)} + c$
$(x - \sec (x)) e^{\sin (x)} + c$
$(x + \tan (x)) e^{\sin (x)} + c$
$(x - \tan (x)) e^{\sin (x)} + c$

37.
If $\int_{0}^{\frac{π}{2}} \frac{dx}{3 \cos (x) 5} = {kcot}^{- 1} (2)$ , then what is the value of k ?

$\frac{1}{4}$

$\frac{1}{2}$

1

2

$\frac{1}{2}$

$I = \int_{0}^{\frac{π}{2}} \frac{d x}{3 \cos (x) + 5} I = \int_{0}^{\frac{π}{2}} \frac{d x}{3 [\frac{1 - \tan^{2} (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})}] + 5} I = \int_{0}^{\frac{π}{2}} \frac{(1 + \tan^{2} (\frac{x}{2})) d x}{3 - 3 \tan^{2} (\frac{x}{2}) + 5 + 5 \tan^{2} (\frac{x}{2})} I = \int_{0}^{\frac{π}{2}} \frac{\sec^{2} (\frac{x}{2}) dx}{2 \tan^{2} (\frac{x}{2}) + 8} I = \frac{1}{2} \int_{0}^{\frac{π}{2}} \frac{\sec^{2} (\frac{x}{2}) dx}{2 \tan^{2} (\frac{x}{2}) + 2^{2}} Put \tan (\frac{x}{2}) = y \Rightarrow \frac{1}{2} \sec^{2} (\frac{x}{2}) dx = y$

$\Rightarrow I = \int_{0}^{1} \frac{dy}{y^{2} + 2^{2}} \Rightarrow I = \frac{1}{2} \tan^{- 1} (\frac{y}{2}) \Rightarrow I = \frac{1}{2} \tan^{- 1} (\frac{1}{2}) - 0 Also I = \frac{1}{2} \tan^{- 1} (\frac{1}{2}) = {kcot}^{- 1} (2) (∵ \tan^{- 1} (x) = {kcot}^{- 1} (\frac{1}{x})) ∴ k = \frac{1}{2}$