Integrate the functions in Exercises: x 1+2x^2 - Vidyadip

Question

Integrate the functions in Exercises:
$\text{x}\sqrt{1+2\text{x}^2}$

✓

Answer

$\text{Let}\text{ I}=\int\text{x}\sqrt{1+2\text{x}^2}\text{ dx} =\frac{1}{4}\int\sqrt{1+2\text{x}^2}(4\text{x dx}) \ \ \ \ \ \ ....\text{(i)} $
Putting $ 1+2\text{x}^2=\text{t}\ \ \ \ \Rightarrow \ \ \ \ \ 4\text{x}=\frac{\text{dt}}{\text{dx}}\ \ \ \ \ \Rightarrow \ \ \ \ 4\text{x}\text{ dx}=\text{dt} $
$\therefore\ \ \ \ \ $From eq. (i), $\text{I}=\frac{1}{4}\int\sqrt{\text{t}}\text{ dt}=\frac{1}{4}\int\text{t}^{\frac{1}{2}}\text{ dt}$
$=\frac{1}{4}\frac{\text{t}^{^3/_2}}{{^3/_2}}+\text{c}=\frac{1}{4}\dot\ \frac{2}{3}\text{t}^{^3/_2 } +\text{c} $
$=\frac{1}{6}\big(1 + 2\text{x}^2\big)^{^3/_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 "1 Marks Question" 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

$\text{If }x \in \text{N and} \begin{bmatrix} \text{x + 3} & -2 \\ \text{-3x} & \text{2x} \\ \end{bmatrix} = 8, $ then find the value of $x.$

One card is drawn at random from a well shuffled deck of 52 cards.
In which of the following cases are the events E and F independent?
E: the card drawn is black
F: the card drawn is a king

Define degree of a differential equation.

Prove that $ \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) + \cos ^ { - 1 } \left( \frac { 12 } { 13 } \right) = \cos ^ { - 1 } \left( \frac { 33 } { 65 } \right).$

Find the direction cosines of the following vectors:
$3\hat{\text{i}}-4\hat{\text{k}}$

Classify the following as scalar and vector quantities.
Force.

Given that the events A and B are such that P(A) = $\frac{1}{2}$, $P(A \cup B)=\frac{3}{5}$ and P(B) = p.
Find p if they are independent.

If matrix $A+B=\left[\begin{array}{ll}2 & 4 \\ 1 & 2\end{array}\right], A-B=\left[\begin{array}{ll}0 & 1 \\ 2 & 2\end{array}\right]$, then find matrix $A$.

Given an example of:
A row matrix which is also a column matrix,

$A = [a_{ij}]_{ m \times n}$ is a square matrix, if