Evaluate the following integrals: 1- 1+ e^ -x 2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\sqrt{1-\sin\text{x}}}{1+\cos\text{x}}\text{e}^{\frac{-\text{x}}{2}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\sqrt{1-\sin\text{x}}}{1+\cos\text{x}}\text{e}^{\frac{-\text{x}}{2}}\text{dx}$
Put $=\frac{\text{x}}{2}=\text{t}$
$\Rightarrow\text{x}=2\text{t}$
$\text{dx}=2\text{dt}$
$\therefore\int\frac{\sqrt{1-\sin\text{x}}}{1+\cos\text{x}}\text{e}^{-\frac{\text{x}}{2}}\text{dx}$
$=2\int\frac{\sqrt{1-\sin2\text{t}}}{1+\cos2\text{t}}\text{e}^{-\text{t}}\text{dt}$ $\big[\because\sin^2\text{t}+\cos^2\text{t}=1\big]$
$=2\int\frac{\sqrt{\sin^2\text{t}+\cos^2\text{t}-2\sin\text{t}\cos\text{t}}}{1+\cos2\text{t}}\text{e}^{-\text{t}}\text{dt}$
$=2\int\frac{\sqrt{(\cos\text{t}-\sin\text{t})^2}}{2\cos^2\text{t}}\text{e}^{-\text{t}}\text{dt}$
$=2\int\frac{(\cos\text{t}-\sin\text{t})}{2\cos^2\text{t}}\text{e}^{-\text{t}}\text{dt}$
$=\int(\sec\text{t}-\tan\text{t}\sec\text{t})\text{e}^{-\text{t}}\text{dt}$
$=\int\sec\text{e}^{-\text{t}}\text{dt}-\int\tan\text{t}\sec\text{e}^{-\text{t}}\text{dt}$
Integrating by parts
$=\text{e}^{-\text{t}}\sec\text{t}+\int\text{e}^{-\text{t}\frac{\text{d}}{\text{dt}}}(\sec\text{t})\text{dt}-\int\tan\text{t}\sec\text{t}\text{ e}^{-\text{t}}\text{dt}$
$=-\text{e}^{-\text{t}}\sec\text{t}+\int\text{e}^{-\text{t}}\sec\text{t}\tan\text{t dt}-\int\sec\text{t}\tan\text{t}\text{ e}^{-\text{}t}\text{dt}$
$=-\text{e}^{-\text{t}}\sec\text{t+C}$
Putting the value of t
$=\text{-e}^{-\frac{\text{x}}{2}}\sec\frac{\text{x}}{2}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following differential equation:
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$

Evaluate the following integrals:$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$

In a large school, $80\%$ of the pupil like Mathematics. A visitor to the school asks each of $4$ pupils, chosen at random, whether they like Mathematics.
(i) Calculate the probabilities of obtaining an answer yes from$ 0, 1, 2, 3, 4$ of the pupils.
(ii) Find the probability that the visitor obtains answer yes from at least$ 2$ pupils:
(a) when the number of pupils questioned remains at $4.$
(b) when the number of pupils questioned is increased to $8.$

Find the area under the curve $\text{y}=\sqrt{6\text{x}+4}$ above x-axis from $x = 0$ to $x = 2$. Draw a sketch of curve also.

Classify the following functions as injection, surjection or bijection:f : R → R, defined by f(x) = 3 - 4x

Solve the following systems of linear equations by cramer's rule:

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{2\text{y}+\text{x}}$

If $\text{x}=\text{a}\sin2\text{t}(1+\cos 2\text{t})$ and $\text{y}=\text{b}\cos\text{t}(1-\cos2\text{t}),$ show that at $\text{t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}\text{ t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}$

Show the solution zone of the following inequalities on a graph paper:
$5\text{x}+\text{y}\geq10$
$\text{x}+\text{y}\geq6$
$\text{x}+4\text{y}\geq12$
$\text{x}\geq,\text{y}\geq0$

A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of A and B, and the number of man-hours the firm has available per week are as follows:

Gadget	Fondry	Machine-shop
A B	10 6	5 4
Firm's capacity per week	1000	600

The profit on the sale of A is Rs. 30 per unit as compared with Rs. 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.