Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5\cos\text{x}+3\sin\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5\cos\text{x}+3\sin\text{x}}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5\Bigg(\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)+3\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}\text{ dx}$ $\Bigg[\because\sin\text{A}=\bigg(\frac{2\tan\frac{\text{A}}{2}}{1+\tan^2\frac{\text{A}}{2}}\bigg),\cos\text{A}=\bigg(\frac{1-\tan^2\frac{\text{A}}{2}}{1+\tan^2\frac{\text{A}}{2}}\bigg)\Bigg]$
$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{1+\tan^2\frac{\text{x}}{2}}{5-5\tan^2\frac{\text{x}}{2}+6\tan\frac{\text{x}}{2}}\text{ dx}$
$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{\sec^2\frac{\text{x}}{2}}{5-5\tan^2\frac{\text{x}}{2}+6\tan\frac{\text{x}}{2}}\text{ dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$ Then, $\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{ dx}=\text{dt}$ Also, $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{2}, \text{t}=1$$\therefore\ \text{I}=\int_{0}^\limits{1}\frac{2\text{ dt}}{5-5\text{t}^2+6\text{t}}$
$\text{I}=\frac{1}{5}\int_{0}^\limits{1}\frac{2\text{dt}}{1-\text{t}^2+\frac{6}{5}\text{t}+\frac{36}{100}-\frac{36}{100}}$
$=\frac{2}{5}\int_{0}^\limits{1}\frac{\text{dt}}{-\big(\text{t}-\frac{6}{10}\big)^2+\frac{136}{100}}$
$=\frac{2}{5}\times\frac{10}{\sqrt{136}}\Bigg[-\log\Bigg(\frac{\text{t}-\frac{6}{10}-\frac{\sqrt{136}}{10}}{\text{t}-\frac{6}{10}+\frac{\sqrt{136}}{10}}\Bigg)\Bigg]^1_0$
$=\frac{1}{\sqrt{34}}\bigg[-\log\bigg(\frac{4-2\sqrt{34}}{4+2\sqrt{34}}\bigg)+\log\bigg(\frac{-6-2\sqrt{34}}{-6+2\sqrt{34}}\bigg)\bigg]$
$=\frac{1}{\sqrt{34}}\log\bigg(\frac{6+2\sqrt{34}}{6-2\sqrt{34}}\times\frac{4+2\sqrt{34}}{4-2\sqrt{34}}\bigg)$
$=\frac{1}{\sqrt{34}}\log\bigg(\frac{160+20\sqrt{34}}{160-20\sqrt{34}}\bigg)$
$=\frac{1}{\sqrt{34}}\log\bigg(\frac{8+\sqrt{34}}{8-\sqrt{34}}\bigg)$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The probability that A hits a target is $\frac{1}{3}$ and the probability that B hits it, is $\frac{2}{5}$, What is the probability that the target will be hit, if each one of A and B shoots at the target?
→
If $A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3\end{array}\right]$, verify that $A(\operatorname{adj} A)=(\operatorname{adj} A) A=|A| \cdot \mid$
→
Find the equation of the plane that contains the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})-4=0$ and $\vec{\text{r}}\cdot(2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})+5=0$ and which is perpendicular to the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+3\hat{\text{j}}-6\hat{\text{k}})+8=0.$
→
If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find $A^{-1}.$ Using $A^{-1},$ solve the system of linear equations:
$x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7$
→
Integrate the following w. r. t. x:

$\frac{12 x^2-2 x-9}{\left(4 x^2-1\right)(x+3)}$

→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}\text{ dx}$
→
Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution.
→
Solve the following differential equation:
$(\text{x}-\text{y})\frac{\text{dy}}{\text{dx}}=\text{x + 2y}$
→
Evaluate the following intregals:
$\int\frac{1}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}\ \text{dx}$
→
Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
→