1- 1+ dx - Vidyadip

Question

$\int\frac{1-\cos\text{x}}{1+\cos\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1-\cos\text{x}}{1+\cos\text{x}}\times\text{dx}.$ Then,
$\text{I}=\int\frac{2\sin^2\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}\times\text{dx}$
$=\int\frac{\sin^2\frac{\text{x}}{2}}{\cos^2\frac{\text{x}}{2}}\times\text{dx}$
$=\int\tan^2\frac{\text{x}}{2}\text{dx}$
$=\int\Big(\sec^2\frac{\text{x}}{2}-1\Big)\text{dx}$
$=\frac{\tan\frac{\text{x}}{2}}{\frac{1}2{}}-\text{x+c}$
$=2\tan\frac{\text{x}}{2}-\text{x+c}$

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 "3 Marks" from Indefinite Integrals→Indefinite 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 $\text{y}=\sqrt{\log\text{x}+\sqrt{\log\text{x}+\sqrt{\log\text{x}+\ .... \text{to }\infty}}},$ prove that $(2\text{y}-1)\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{x}}$

Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibits 4.

Solve system of linear equations, using matrix method.

Let A = [-1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x²

Evaluate the following integrals:

$\int\frac{\sin8\text{x}}{\sqrt{9+\sin^44\text{x}}}\text{ dx}$

Find the perpendicular distence of the point (3, -1, 11) from the line $\frac{\text{x}}{2}=\frac{\text{y}-2}{-3}=\frac{\text{z}-3}{4}.$

For the principal values, evaluate the following:
$\tan^{-1}\big(\sqrt3\big)-\sec^{-1}(-2)$

Show that $\cos\Big(2\tan^{-1}\frac{1}{7}\Big)=\sin\Big(4\tan^{-1}\frac{1}{3}\Big).$

Find a unit vector in the direction of $\overrightarrow{\text{PQ}},$ where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively.

Evaluate the following integrals:
$\int\cot^3\text{x }\text{cosec}^2\text{x}\text{ dx}$