$\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}\sqrt{\mathit{t}\mathit{a}\mathit{n}\mathit{x}}$ + $\sqrt{\mathit{c}\mathit{o}\mathit{t}\mathit{x}}$ dx
= $\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}$ $\left(\frac{\sqrt{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}{\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}+\frac{\sqrt{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}{\sqrt{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}\right)$ dx
= $\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}$ $\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\sqrt{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}\right)$ dx
= $\sqrt{2}\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}$ $\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\sqrt{2\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}\right)$ dx
$\sqrt{2}\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}$ $\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\sqrt{1-\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{c}\mathit{o}\mathit{s}{\mathit{x}}^2}}\right)$ dx
Put sin x - cos x = t ⇒ (cos x + sin x) dx = dt
If x = 0 , t = 0 - 1 = - 1
and if x = $\frac{\mathit{\pi }}{4}$ , t = $\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}$ = 0
∴ $\underset{0}{\overset{\frac{\mathit{\pi }}{4}}{\int }}$ $\sqrt{\mathit{t}\mathit{a}\mathit{n}\mathit{x}}+\sqrt{\mathit{c}\mathit{o}\mathit{t}\mathit{x}}$ dx = $\sqrt{2}\underset{-1}{\overset{0}{\int }}\frac{\mathit{d}\mathit{t}}{\sqrt{1-{\mathit{t}}^2}}$
= $\sqrt{2}|\mathit{s}\mathit{i}{\mathit{n}}^{-1}\mathit{t}{|}_{-1}^0$
= $\sqrt{2}|\mathit{s}\mathit{i}{\mathit{n}}^{-1}0-\mathit{s}\mathit{i}{\mathit{n}}^{-1}\left(-1\right)|$
= $\sqrt{2}|0+\frac{\mathit{\pi }}{2}|$
= $\sqrt{2}\times \frac{\mathit{\pi }}{2}$
OR
$\underset{1}{\overset{3}{\int }}2{\mathit{x}}^2$ + 5x dx
Here, a = 1 , b = 3 , f (x) = $2{\mathit{x}}^2$ + 5x
∴ nh = b - a = 3 - 1 = 2
Now $\underset{\mathit{a}}{\overset{\mathit{b}}{\int }}$ f (x) dx = $\underset{\mathit{h}\to 0}{\lim }$ f (a) + f (a + h) + f (a + 2h) + ... + f (a + (n - 1)h)
∴ $\underset{1}{\overset{3}{\int }}2{\mathit{x}}^2$ + 5x dx
= $\underset{\mathit{h}\to 0}{\lim }$ h |2 ${\left(1\right)}^2$ +5 (1) + 2 ${\left(1+\mathit{h}\right)}^2$ + 5 (1 + h) + [2${\left(1+2\mathit{h}\right)}^2$ + 5 (1 + 2h)] ... + [2 ${\left(1+\left(\mathit{n}-1\right)\mathit{h}\right)}^2$ + 5 (1 + (n - 1) h)]|
= $\underset{\mathit{h}\to 0}{\lim }$ |7 + ($2{\mathit{h}}^2$ + 9h + 7) + ($8{\mathit{h}}^2$ + 18h + 7) + ... + (2 ${\left(\mathit{n}-1\right)}^2{\mathit{h}}^2$ + 9 (n - 1) h + 7)|
= $\underset{\mathit{h}\to 0}{\lim }$ |7n + $2{\mathit{h}}^2$ ($1^2+2+...+{\left(\mathit{n}-1\right)}^2$) + 9h (1 + 2 + ... + (n - 1))|
= $\mathit{l}\mathit{i}\underset{\mathit{h}\to 0}{\mathit{m}}$
= $\underset{\mathit{h}\to 0}{\lim }$
= $\underset{\mathit{h}\to 0}{\lim }$
= 14 + $\frac{16}{3}$ + 18 = $\frac{112}{3}$