Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{2+\sin\text{x}+\cos\text{x}}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{2+\sin\text{x}+\cos\text{x}}\text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}},\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=-\int\frac{1}{2+\begin{pmatrix}\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\end{pmatrix}+\begin{pmatrix}\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\end{pmatrix}}\text{dx}$
$=\int\frac{\begin{pmatrix}1+\tan^2\frac{\text{x}}{2}\end{pmatrix}}{2+2\tan^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}}\text{dx}$
$=\int\frac{\begin{pmatrix}\sec^2\frac{\text{x}}{2}\end{pmatrix}}{\tan^2\frac{\text{x}}{2}+\tan\frac{\text{x}}{2}+3}\text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$\text{I}=\int\frac{2\text{dt}}{\text{t}^2+2\text{t}+3}$
$\text{I}=2\int\frac{\text{dt}}{\text{t}^2+2\text{t}+1-1+3}$
$\text{I}=2\int\frac{\text{dt}}{(\text{t}+1)^2+(\sqrt2)^2}$
$=\frac{2}{\sqrt2}\tan^{-1}\Big(\frac{\text{t}+1}{\sqrt2}\big)+\text{C}$
$\text{I}=\sqrt{2}\tan^{-1}\Big(\frac{\tan\frac{\text{x}}{2}+1}{\sqrt2}\Big)+\text{C}$

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 "4 Marks" 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

If $\cos ^{-1} \frac{x}{a}+\cos ^{-1} \frac{y}{b}=\alpha$ then prove that$\frac{x^2}{a^2}-\frac{2 x y}{a b} \cos \alpha+\frac{y^2}{b^2}=\sin ^2 \alpha$
→
Evaluate the following intregals:
$\int\frac{\text{x}+1}{\sqrt{4+5\text{x}-\text{x}}}\text{ dx}$
→
Solve the following differential equation
$\text{x}(\text{x}^{2} - 1)\frac{\text{dy}}{\text{dx}} = 1, \text{y}(2) = 0$
→
Differentiate the following functions with respect to x:
$\text{e}^\text{x}\log\sin2\text{x}$
→
Evaluate the following integrals:
$\int\frac{1}{\cos\text{x}(5-4\sin\text{x})}\ \text{dx}$
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}\text{a}&\text{h}&\text{g}\\\text{h}&\text{b}&\text{f}\\\text{g}&\text{f}&\text{c} \end{vmatrix}$
→
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
→
Differentiatet $\tan^{-1}\Big(\frac{x}{\sqrt{1-x^2}}\Big)$ with respect to $\sin^{-1}(2x\sqrt{1-x^2}).$
→
Find the vector equation of the line through the origin which is perpendicular to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=3$
→
If the product of the distances of the point (1, 1, 1) from the origin and the plane x - y + z + λ = 0 be 5, find the value of λ.
→