Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Evaluate the following integrals:
$\int\frac{1}{4\text{x}^2+12\text{x}+5}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{4\text{x}^2+12\text{x}+5}\text{dx}$
$=\frac{1}{4}\int\frac{1}{\text{x}^2+3\text{x}+\frac{5}{4}}\text{dx}$
$=\frac{1}4{}\int\frac{1}{\text{x}^2+2\times\text{x}\times\big(\frac{3}{2}\big)+\big(\frac{3}{2}\big)^2-\big(\frac{3}{2}\big)^2+\frac{5}{4}}\text{dx}$
$\text{I}=\frac{1}{4}\int\frac{1}{\Big(\text{x}+\frac{3}{2}\Big)^2-1}\text{dx}$
Let $\Big(\text{x}+\frac{3}{2}\Big)=\text{t}\ \dots(1)$
$\Rightarrow\text{dx = dt}$
So,
$\text{I}=\frac{1}{4}\int\frac{1}{\text{t}^2-(1)^2}\text{dt}$
$\text{I}=\frac{1}{4}\times\frac{1}{2\times(1)}\log\bigg|\frac{\text{t}-1}{\text{t}+1}\bigg|+\text{C}$ $\Big[\text{Since,} \int\frac{1}{\text{x}^2-\text{a}^2}\text{dx}=\frac{1}{2\text{a}}\log\bigg|\frac{\text{x}-\text{a}}{\text{x+a}}\bigg|+\text{C}\Big]$
$\text{I}=\frac{1}{8}\log\Bigg|\frac{\text{x}+\frac{3}{2}-1}{\text{x}+\frac{3}{2}+1}\Bigg|+\text{C}$ [using (1)]
$\text{I}=\frac{1}{8}\log\bigg|\frac{2\text{x}+1}{2\text{x}+5}\bigg|+\text{C}$

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 "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

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{x}}{1+6\text{x}^3}\Big)$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=​​\begin{cases}-2,&\text{if }\text{ x}\leq-1\\2\text{x},&\text{if }-1<\text{x}\leq1\\2,&\text{if }\text{ x}>1\end{cases}$
→
$\int\text{x}(1-\text{x})^{23}\text{dx}$
→
Find the shortest distance between the lines $\text{x + 1 = 2y = -12z and x = y + 2 = 6z - 6}.$
→
$\int\frac{\text{x}^2+3\text{x}-1}{(\text{x}+1)^2}\text{dx}$
→
If $\text{A} = \begin{bmatrix}2&-3&5\\3&2&-4\\1&1&-2\end{bmatrix},$ find $A^{-1}.$ Using $A^{-1}$ solve the system of equations:
$2x - 3y + 5z = 11$
$3x + 2y - 4z = -5$
$x + y - 2z = -3$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\text{x}^{11}-3\text{x}^9+5\text{x}^7-\text{x}^5+1}{\cos^2\text{x}}\text{ dx}$
→
Examine the differentiability of f, where f is defined by:
$\text{f(x)}=\begin{cases}1+\text{x},&\text{if x}\leq2\\5-\text{x},&\text{if x}>2\end{cases}$
at x = 2.
→
If $\text{A}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0&-1\\1&0\end{bmatrix},$ then show that $(\text{A}+\text{B})(\text{A}-\text{B})\neq\text{A}^2-\text{B}^2.$
→
If $\text{x}=\sin^{-1}\Big(\frac{2\text{t}}{1+\text{t}^2}\Big)$ and $\text{y}=\tan^{-1}\Big(\frac{2\text{t}}{1-\text{t}^2}\Big),-1<\text{t}<1,$ prove that $\frac{\text{dy}}{\text{dx}}=1$
→