Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{\cos\text{x}}{\cos3\text{x}}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\cos\text{x}}{\cos3\text{x}}\ \text{dx}$
$=\int\frac{\cos\text{x}}{(4\cos^3\text{x}-3\cos\text{x})}\ \text{dx}$ $\big[\cos3\text{A}=4\cos^3\text{A}-3\cos\text{A}\big]$
$=\int\frac{1}{4\cos^2\text{x}-3}\ \text{dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\Rightarrow\text{I}=\int\frac{\sec^2\text{x}}{4-3\sec^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{4-3(1+\tan^2\text{x})}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{1-3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{1-(\sqrt{3}\tan\text{x})^2}\ \text{dx}$
Let $\sqrt{3}\tan\text{x}=\text{t}$
$\Rightarrow\sqrt{3}\sec^2\text{x dx}=\text{dt}$
$\Rightarrow\sec^2\text{x}\text{ dx}=\frac{\text{dt}}{\sqrt{3}}$
$\therefore\text{I}=\frac{1}{\sqrt{3}}\int\frac{\text{dt}}{1^2-\text{t}^2}$
$=\frac{1}{\sqrt{3}}\times\frac{1}{2}\ln\big|\frac{1+\text{t}}{1-\text{t}}\big|+\text{C}$
$=\frac{1}{2\sqrt{3}}\ln\Big|\frac{1+\sqrt{3}\tan\text{x}}{1-\sqrt{3}\tan\text{x}}\Big|+\text{C}$

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 "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

Find a point on the parabola $y = (x - 4)^2$​​​​​​​, where the tangent is parallel to the chord joining $(4, 0)$ and $(5, 1).$
→
If $\text{x}-\text{e}^{\tan\text{x}}+\sqrt{\frac{\text{x}^2+1}{2}},$ find $\frac{\text{dy}}{\text{dx}}$
→
Find the shortest distance between the lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$
→
Show that the function g(x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
→
Solve the following equation for x:
$\tan^{-1}\frac{\text{x}-2}{\text{x}-1}+\tan^{-1}\frac{\text{x}+2}{\text{x}+1}=\frac{\pi}{4}$
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin(3\text{x}+1)\text{dx}$
→
Classify the following functions as injection, surjection or bijection:$f : R \rightarrow R$, defined by$ f(x) = 5x^3 + 4$
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\text{x}^{11}-3\text{x}^9+5\text{x}^7-\text{x}^5+1}{\cos^2\text{x}}\text{ dx}$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 2 & -1 & 4 \\ 4 & 0 & 7 \\ 3 & -2 & 7 \end{bmatrix}$
→
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2\sqrt{2}}}\Big)$
→