Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$
Let $2-\text{x}=\text{t}$
$-\text{dx}=\text{dt}$
$\text{dx}=-\text{dt}$
So, $\text{I}=-\int\frac{1}{\sqrt{\text{t}^2+(1)^2}}\text{ dt}$
$\text{I}=-\log\big|\text{t}+\sqrt{\text{t}^2+1}\big|+\text{C}$ $\Big[$Since $\int\frac{1}{\sqrt{\text{x}^2+\text{a}^2}}\text{ dx}=\log\big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\big|+\text{C}\Big]$
$\text{I}=-\log\big|(2-\text{x})+\sqrt{(2-\text{x})^2+1}\big|+\text{C}$ $[$Since $\text{t}=(2-\text{x})]$

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 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

$\text{Find}: \int(x + 3) \sqrt{3 - 4x - x^{2}} dx.$
→
Explain if Rolle's theorem is applicable to any one of the following functions.
  1. $\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[5,9]$
  2. $\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[-2,2]$
Can you say something about the converse of Rolle's Theorem from these functions?
→
Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result:
$\text{A}=\begin{bmatrix}3 & -2 &-4\\3 & -2&-5\\-1&-1& 2\end{bmatrix}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
→
Construct $a_{2 \times 2}$​​​​​​​ matrix, where,
  1. $\text{a}_\text{ij}=\frac{(\text{i}-2\text{j})^2}{2}$
  2. $\text{a}_\text{ij}=|-2\text{i}+3\text{j}|$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^\text{n}\text{x}}{\sin^\text{n}\text{x}+\cos^\text{n}\text{x}}\text{ dx}$
→
The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase
  1. In total surface area, and
  2. In the volume, assuming that k is small?
→
Using the proprties of determinants in Exercise:
$\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}=4\text{xyz}$
→
A factory owner purchases two types of machines, $A$ and $B,$ for his factory. The requirements and limitations for the machines are as follows:
 
Area occupied by the
machine
Labour force for each
machine
Daliy outputin
units
Machines
$1000$ sp.m
$12$ mem
$60$
Machines
$1200$ sp.m
$8$ mem
$40$
He has an area of $7600$ sq. m available and $72$ skilled men who can operate the machines.
How many machines of each type should he buy to maximize the daily output?
→
$\text{Let A = Q} \times \text{Q}$ and let $*$ be a binary operation on A defined by$\text{(a, b)} $*$ \text{(c, d) = (ac, b + ad)} \text{ for (a, b), (c, d)} \in \text{A}.$ Determine, whether $*$ is commutative and associative. Then, with respect to $*$ on A.
  1. Find the identity element in A.
  2. Find the invertible elements of A.
→