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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)

2f( - 1)

$f' (\frac{1}{x}) + x^{3} f' (x) = 0 . f' (\frac{1}{x}) = - x^{3} f' (x) - \frac{1}{x^{2}} f' (\frac{1}{x}) = xf' (x) \int_{- 1}^{1} (- \frac{1}{x^{2}}) f' (\frac{1}{x}) = \int_{- 1}^{1} xf' (x) Let \frac{1}{x} = t x = - 1, t = 1 - \frac{1}{x^{2}} dx = dt x = 1, t = 1 \int_{- 1}^{1} f' (t) dt = \int_{- 1}^{1} xf' (x) d x {[f (t)]}_{- 1}^{1} = x \int_{- 1}^{1} f' (x) dx - \int_{- 1}^{1} 1 (\int f' (x)) dx = {[xf (x)]}_{- 1}^{1} - \int_{- 1}^{1} f (x) dx \Rightarrow f (1) - f (- 1) = [1 f (1) - (- 1) f (- 1)] - \int_{- 1}^{1} f (x) dx \Rightarrow f (1) - f (- 1) = f (1) + f (- 1) - \int_{- 1}^{1} f (x) dx \Rightarrow \int_{- 1}^{1} f (x) dx = 2 f (- 1) \Rightarrow \int f' (\frac{1}{x}) dx = f (x)$

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

38.

What is $\int_{1}^{3} |1 - x^{4}| d x = ?$

$- \frac{232}{5}$
$- \frac{116}{5}$
$\frac{116}{5}$
$\frac{232}{5}$

39.

$Consider the integral I_{1} = \int_{0}^{π} \frac{x}{1 + \sin (x)} d x and I_{2} = \int_{0}^{π} \frac{(π - x) dx}{1 - \sin (π + x)}$

What is the value of I₁ + I₂ ?

2 $π$
$π$
$\frac{π}{2}$
0

40.