Evaluate the following integrals: _ 0 ^1 2x 1+x^4 dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{2\text{x}}{1+\text{x}^4}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{2\text{x}}{1+\text{x}^4}\text{ dx}$
Let $\text{x}^2=\text{t}$ Then, $2\text{xdx}=\text{dt}$
When $\text{x}=10,\text{t}=0$ and $\text{x}=1,\text{t}=1$
$\therefore\ \text{I}=\int_{0}^\limits{1}\frac{2\text{x}}{1+\text{x}^4}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{1}\frac{1}{1+\text{t}^2}\text{ dt}$
$\Rightarrow\text{I}=\big[\tan^{-1}\text{t}\big]^1_0$
$\Rightarrow\text{I}=\tan^{-1}1-\tan^{-1}0$
$\Rightarrow\text{I}=\frac{\pi}{4}$

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 "3 Marks Question" 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

Determine P(E|F) in Exercises.
Two coins are tossed once, where
E : tail appears on one coin, F : one coin shows head.

Consider the binary operation $*$ and o defined by the following tables on set $S = \{a, b, c, d\}.$

$o$	$a$	$b$	$c$	$d$
$a$	$a$	$a$	$a$	$a$
$b$	$a$	$b$	$c$	$d$
$c$	$a$	$c$	$d$	$b$
$d$	$a$	$d$	$b$	$c$

Solve the Linear Programming Problem graphically:
Maximise Z = 3x + 4y subject to the constraints: $x + y \le4, \ x \geq 0, \ y \geq 0$

If $\text{A}=\begin{bmatrix}2&3\\5&7\end{bmatrix},\text{ B}=\begin{bmatrix}-1&0&2\\3&4&1\end{bmatrix},\text{C}=\begin{bmatrix}-1&2&3\\2&1&0\end{bmatrix},$ find2B + 3A and 3C - 4B.

Write a value of $\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$

If $\text{y}=\sqrt{\text{x}+\sqrt{\text{x}+\sqrt{\text{x}+\ .... \text{to }\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{2\text{y}-1}$

A circular disc of radius 3cm is being heated. Due to expansion, its radius increases at the rate of 0.05cm/ sec. Find the rate at which its area is increasing when radius is 3.2cm.

Prove that the given vectors are coplanar:
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $-\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$

Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.

Integrate the function $\sqrt{x^{2}+4 x+1}$