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61.

The value of integral $\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{(\sin (x) - xcos (x))}{x (x + \sin (x))} d x$

Fis increasing in (0, $\frac{π}{2}$ ) and $(\frac{3 π}{2}, 2 π)$ and decreasing in ( $\frac{π}{2}, \frac{3 π}{2}$ ).

Given function is,

F(x) = $\int_{0}^{x} \frac{\cos (t)}{(1 + t^{2})} d t, 0 \leq x \leq 2 π$

On differentiation w.r.t. x., (apply Leibnitz rule)

F'(x) = $\frac{\cos (x)}{1 + x^{2}} \times 1 = \frac{\cos (x)}{1 + x^{2}}, where (1 + x^{2}) > 0$

Here, $\cos (x) > 0 \Rightarrow x \in (0, \frac{π}{2}) \cup (\frac{3 π}{2}, 2 π)$

and $\cos (x) < 0 \Rightarrow x \in (\frac{π}{2}, \frac{3 π}{2})$

So, F is increasing in $(0, \frac{π}{2})$ and $(\frac{3 π}{2}, 2 π)$ and decreasing in $(\frac{π}{2}, \frac{3 π}{2})$

63.

The value of the integral $\int_{\frac{π}{6}}^{\frac{π}{2}} (\frac{1 + \sin (2 x) + \cos (2 x)}{\sin (x) + \cos (x)}) d x$ is equal to

64.

The value of the integral $\int_{0}^{\frac{π}{2}} \frac{1}{1 + {(\tan (x))}^{101}} d x$ is equal to

65.

The integrating factor of the differential equation

$3 {xlog}_{e} (x) \frac{d y}{d x} + y = 2 \log_{e} (x)$ is given by

66.

The value of the integral $\int_{0}^{\frac{π}{4}} \frac{\sin (x) + \cos (x)}{3 + \sin (2 x)} d x$ is equal to

67.

The value of the integral $\int_{- 2}^{2} (1 + 2 \sin (x)) e^{|x|} d x$ is equal to

68.

The value of the integral $\int_{1}^{5} [|x - 3| + |1 - x|] d x$ is equal to

69.

Let [x] denote the greatest integer less than or equal to x, then the value of the integral $\int_{- 1}^{1} (|x| - 2 [x]) d x$ is equal to

70.

Prove that