Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
Evaluate the following integrals:$\int_{0}^\limits{1}\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}\text{ dx}$
✓

Answer

Let $\text{e}^\text{x}=\text{t}$ Then, $\text{e}^\text{x}\text{ dx}=\text{dt}$ When $\text{e}^\text{x}=0,\text{t}=1$ and $\text{x}=1,\text{t}=\text{e}$$\therefore\ \text{I}=\int_{0}^\limits{1}\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}\text{ dx}$
$\Rightarrow\text{I}=\int_{1}^\limits{\text{e}}\frac{\text{dt}}{1+\text{t}^{2}}$
$\Rightarrow\text{I}=\big[\tan^{-1}\text{x}\big]^\text{e}_1$
$\Rightarrow\text{I}=\tan^{-1}\text{e}-\tan^{-1}1$
$\Rightarrow\text{I}=\tan^{-1}\text{e}-\frac{\pi}{4}$

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.(3 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Diffrentiate the following w.r.t.x

$\frac{\left(x^2+2\right)^4}{\sqrt{x^2+5}}$

→
Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$
→
Prove by vector method that the angle subtended on semicircle is a right angle.
→
If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find $\text{P}(\text{X}\geq4).$
→
Evaluate :

$\int_0^3 x^2(3-x)^{\frac{5}{2}} \cdot d x$

→
A die is tossed twice. Find the probability of getting a number greater than 3 on each toss.
→
If $A=\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 1 \\ 1 & 1 & 1\end{array}\right]$, then reduce it to $I_3$ by using row transformations.
→
Construct the composition table for $\times _4$ on set $S = {0, 1, 2, 3}$.
→
If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=5$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=8,$ find $\vec{\text{a}}.\vec{\text{b}}.$
→
Evaluate the following integrals:$\int\frac{\cos\text{x}}{\sqrt{\sin^2\text{x}-2\sin\text{x}-3}}\text{ dx}$
→