Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{1-\sin\text{x}+\cos\text{x}}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{1-\sin\text{x}+\cos\text{x}}\ \text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}},\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=\int\frac{1}{1-\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}+\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}}$
$=\int\frac{1+\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}-2\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{2-2\tan\frac{\text{x}}{2}}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$=\frac{2}{2}\int\frac{\text{dt}}{1-\text{t}}$
$=-\log|1-\text{t}|+\text{C}$
$\text{I}=-\log\big|1-\tan\frac{\text{x}}{2}\big|+\text{C}$

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

Evaluate the following integrals:
$\int\frac{\sqrt{\tan\text{x}}}{\sin\text{x}\cos\text{x}}\text{dx}$
→
If $\text{y}=\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}},$ prove that $2\text{x}\frac{\text{dy}}{\text{dx}}=\sqrt{\text{x}}-\frac{1}{\sqrt{\text{x}}}$
→
Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0,\text{y}(2)=\text{x}$
→
Maximize : z = 5x + 2y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
→
Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:
$2\text{x}+4\text{y}\leq8$
$3\text{x}+\text{y}\leq6$
$\text{x}+\text{y}\leq4$
$\text{x}\geq0,\text{y}\geq0$
→
Find the cartesian and vector equations of a line which passes through the point ($1, 2, 3$) and is parallel to the line $\frac{-\text{x}-2}{1}=\frac{\text{y}+3}{7}=\frac{2\text{z}-6}{3}.$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \tan(\text{x}+\text{y})$
→
Evaluate the following integrals:$\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$
→
Find the length and the foot ofo perpendicular from the point $\Big(1,\frac{3}{2},2\Big)$ to the plane 2x - 2y + 4z + 5 = 0
→
Evaluate the following integrals:
$\int\frac{\text{x}^2-1}{\text{x}^4+1}\ \text{dx}$
→