Evaluate the following integrals: ^ -1 ( 1-x^2 1+x^2 )dx - Vidyadip

Question

Evaluate the following integrals:

$\int\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)\text{dx}$

✓

Answer

Let $\text{I}=\int\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)\text{dx}$
Let $\text{x}=\tan\text{t}$
$\text{dx}=\sec^2\text{t dt}$
$\text{I}=\int\cos^{-1}\Big(\frac{1-\tan^2\text{t}}{1+\tan^2\text{t}}\Big)\sec^2\text{t dt}$
$=\int\cos^{-1}(\cos2\text{t})\sec^2\text{t dt}$
$=\int2\text{t}\sec^2\text{x dx}$
$=2\Big[\text{t}\int\sec^2\text{t dt}-\int(1\int\sec^2\text{t dt})\text{dt}\Big]$
$=2[\text{t}\tan^2\text{t}-\int\tan\text{t dt}]$
$=2[\text{t}\tan^2\text{t}-\log\sec\text{t}]+\text{C}$
$=2\Big[\text{x}\tan^{-1}\text{x}-\log\sqrt{1+\text{x}^2}\Big]+\text{C}$
$\text{I}=2\text{x}\tan^{-1}\text{x}-\log|1+\text{x}^2|+\text{C}$

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 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

Solve the following differential equations:

$\frac{\text{dy}}{\text{dx}}=\frac{\text{e}^{\text{x}}(\sin^2\text{x}+\sin2\text{x})}{\text{y}(2\log\text{y}+1)}$

Find the general solution of the differential equation
y dx – (x + 2y²) dy = 0.

Differentiate the following functions with respect to x:
$\big(\sin^{-1}\text{x}^4\big)^4$

For each of the differential equation given in find the general solution:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2\log\text{x}$

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sin^3\text{x}\text{ dx}$

Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}\Big)$ and the plane $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\Big)=5.$

If for function $\phi(\text{x})=\lambda\text{x}^2+7\text{x}-4, \phi(5)=97,$ find $\lambda.$

Evaluate the following integrals:
$\int\text{cosec x}\log({\text{cosec x}-\cot\text{x})}\text{dx}$

Find the shortest distance between the lines
$\vec{\text{r}}=\Big(4\hat{\text{i}}-\hat{\text{j}}\Big)+\lambda\Big(\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)$ $\text{and}\ \vec{\text{r}}=\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)+\mu\Big(2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}\Big).$

If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$