Evaluate the following integrals: ^1_0 2x 1+x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^1_0\frac{2\text{x}}{1+\text{x}^2}\text{ dx}$

✓

Answer

We have,
$\text{I}=\int\limits^1_0\frac{2\text{x}}{1+\text{x}^2}\text{ dx}$
Putting $1+\text{x}^2=\text{t}$
$2\text{x dx}=\text{dt}$
When $\text{x}\rightarrow0;\text{ t}\rightarrow1$
And $\text{x}\rightarrow1;\text{ t}\rightarrow2$
$\therefore\ \text{I}=\int\limits^1_0\frac{\text{dt}}{\text{t}^2}$
$=\Big[\log_{\text{e}}|\text{t}|\Big]^2_1$
$=\log_{\text{e}}2-\log_{\text{e}}1$
$=\log_{\text{e}}2-0$
$=\log_{\text{e}}2$

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 "2 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=-4\hat{\text{i}}-6\hat{\text{j}}-2\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+4\hat{\text{j}}+3\hat{\text{k}},\vec{\text{c}}=-8\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$

Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = - 1 and x = 1

A coin is tossed three times. Find the mean for number of heads.

If sinx is an integrating factor of the differential equation $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ than write the value of P.

Write tha value of $\int\sec\text{x}(\sec\text{x}+\tan\text{x})\text{dx}$

Find the domain of definition of $\text{f(x)}=\cos^{-1}\big(\text{x}^2-4\big).$

Find the sum of the vectors $\vec{a}=\hat{i}-2\hat{j}+\hat{k,}\ \ \vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}\ \text{and}\ \vec{c}= \hat{i}-6\hat{j}-7\hat{k}.$

Define an equivalence relation.

A and B are two events such that P (A) ≠ 0. Find P(B|A), if

A is a subset of B
$\text{A}\cap\text{B}=\phi$

Verify that the function $y=e^{x}(a \cos x+b \sin x)$ (implicit or explicit) is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$