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

Question

Evaluate the following integrals:
$\int\limits^1_{-1}\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)\text{dx}$

✓

Answer

Let $\text{I}=\int\limits^1_{-1}\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)\text{dx}$
Here $\text{f(x)}=\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)$
$\text{f}(-\text{x})=\log\Big(\frac{2+\text{x}}{2-\text{x}}\Big)$
$=-\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)$
$=-\text{f(x)}$
Hence f(x) is an odd function,
Therefore,
$\text{I}=0$

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 "5 Marks Questions" from Definite Integrals→Definite 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 equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{2\text{y}+\text{x}}$

$\text{If x = a} (\cos 2\text{t +2t}\sin \text{2t}) \text{and y = a} (\sin \text{2t - 2t}\cos\text{2t}),\text{then find}\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}}.$

Solve the following system of equations by matrix method:
$x - y + z = 2$
$2x - y = 0$
$2y - z = 1$

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\frac{1}{5\cos\text{x}+3\sin\text{x}}\text{ dx}$

If $\text{y}=\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big),$, find $\frac{\text{dy}}{\text{dx}}.$

Evaluate the following integrals:$\int\frac{\text{x}+1}{\text{x}^2+\text{x}+3}\text{ dx}$

A firm manufactures two products $A$ and $B$. Each product is processed on two machines $M_1$ and $M_2$. Product $A$ requires $4$ minutes of processing time on $M_1$ and $8$ min. on $M_2;$ product $B$ requires $4$ minutes on $M_1$ and $4$ min. on $M_2$. The machine $M_1$ is available for not more than $8$ hrs $20$ min. while machine $M_2$ is available for $10$ hrs. during any working day. The products $A$ and $B$ are sold at a profit of Rs. $3$ and Rs. $4$ respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Find a unit vector perpendicular to the plane containing the vectors $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}.}$

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\sin\pi\text{x}\big|\text{dx}$

Solve the following differential equation:
$(\text{x}^2+\text{y}^2)\frac{\text{dy}}{\text{dx}}=8\text{x}^2-3\text{xy}+2\text{y}^2$