Evaluate the following definite integrals: _ - 4 ^ 4 1 1+ dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$ Then,
$\text{I}=\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\times\frac{1-\sin\text{x}}{1-\sin\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{-\frac{\pi}{4}}^\limits{\pi}\frac{1-\sin\text{x}}{1-\sin^2\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{-\frac{\pi}{4}}^\limits{\pi}\frac{1-\sin\text{x}}{\cos^2\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{-\frac{\pi}{4}}^\limits{\pi}\Big(\frac{1}{\cos^2\text{x}}-\frac{\sin\text{x}}{\cos^2\text{x}}\Big)\text{dx}$
$\Rightarrow\text{I}=\int_{-\frac{\pi}{4}}^\limits{\pi}\big(\sec^2\text{x}-\sec\text{x}\tan\text{x}\big)\text{ dx}$
$\Rightarrow\text{I}=\big[\tan\text{x}-\sec\text{x}\big]^\frac{\pi}{4}_{-\frac{\pi}{4}}$
$\Rightarrow\text{I}=\big(1-\sqrt{2}\big)-\big(-1-\sqrt{2}\big)$
$\Rightarrow\text{I}=2$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following differential equation:
$\text{y}^2\text{dx}+(\text{x}^2-\text{xy}+\text{y}^2)\text{dy}=0$

Evaluate: $\int\frac{\text{dx}}{\text{x}\text{(x}^{5}\text{+3)}}$

Evaluate the following definite integrals:
$\int\limits_{0}^{1}\sqrt{\text{x}(1-\text{x})}\text{ dx}$

verify that $\text{y}=\text{e}^{\text{m}\cos^{-1}}$ is a solution of the differential equation $(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{m}^2\text{y}=0$

There are 3 families A, B and C. The number of men, women and children in these families are as under:

	Men	Women	Children
Family A	2	3	1
Family B	2	1	3
Family C	4	2	6

Impact does more children in the family create on matrix multiplication, calculate the daily expenses of each family. What respectively. Only men and women earn and children do not. Using Daily expenses of men, women and children are ₹ 200, ₹ 150 and ₹ 200thesociety?

Using properties of determinants show that $\begin{vmatrix} 1 & 1 & \text{1 + x} \\ 1 & \text{1 + y} & 1 \\ \text{1 + z} & 1 & 1 \end{vmatrix} = \text{xyz + yz + zx + xy}.$

If $\text{u}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ and $\text{v}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big),$ where -1 < x < 1 then write the value of $\frac{\text{dy}}{\text{dx}}.$

Find A and B so that $\text{y}=\text{A}\sin3\text{x}+\text{B}\cos3\text{x}$ satisfy the equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\frac{\text{dy}}{\text{dx}}+3\text{y}=10\cos3\text{x}.$

Evaluate the following integrals:
$\int\limits_{0}^{\pi}\frac{1}{3+2\sin\text{x}+\cos\text{x}}\text{ dx}$

The two adjecent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}.$ Find the unit vector parallel to one of its diagonals. Also, find its area.