Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluvate the following intregals:
$\int\frac{5\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+10}}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{5\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+10}}\ \text{dx}$
Consider,
$5\text{x}+3=\text{A}\frac{\text{d}}{\text{dx}}(\text{x}^2+4\text{x}+10)+\text{B}$
$\Rightarrow5\text{x}+3=\text{A}(2\text{x}+4)+\text{B}$
$\Rightarrow5\text{x}+3=(2\text{A})\text{x}+4\text{A}+\text{B}$
Equating coefficient of like terms
$2\text{A}=5$
$\Rightarrow\text{A}=\frac{5}{2}$
And
$4\text{A}+\text{B}=3$
$\Rightarrow4\times\frac{5}{2}+\text{B}=3$
$\Rightarrow\text{B}=-7$
$\therefore\text{I}=\frac{5}{2}\int\frac{(2\text{x}+4)\text{dx}}{\sqrt{\text{x}^2+4\text{x}+10}}-7\int\frac{\text{dx}}{\sqrt{\text{x}^2+4\text{x}+10}}$
$=\frac{5}{2}\int\frac{(2\text{x}+4)\text{dx}}{\sqrt{\text{x}^2+4\text{x}+10}}-7\int\frac{\text{dx}}{\sqrt{\text{x}^2+4\text{x}+4-4+10}}$
$=\frac{5}{2}\int\frac{(2\text{x}+4)\text{dx}}{\sqrt{\text{x}^2+4\text{x}+10}}-7\int\frac{\text{dx}}{\sqrt{(\text{x}+2})^2+(\sqrt{6})^2}$
Putting, $\text{x}^2+4\text{x}+10=\text{t}$
$\Rightarrow(2\text{x}+4)\text{dx}=\text{dt}$
$\text{I}=\frac{5}{2}\int\frac{\text{dt}}{\sqrt{\text{t}}}-7\log\big|(\text{x}+2)^2+6\big|+\text{C}$
$=\frac{5}{2}\int\text{t}^{-\frac{1}{2}}\text{dt}-7\log\big|\text{x}+2+\sqrt{\text{x}^2+4\text{x}+10}\big|+\text{C}$
$=\frac{5}{2}\times2\sqrt{\text{t}}-7\log\big|\text{x}+2+\sqrt{\text{x}^2+4\text{x}+10}\big|+\text{C}$
$=5\sqrt{\text{x}^2+4\text{x}+10}-7\log\big|\text{x}+2+\sqrt{\text{x}^2+4\text{x}+10}\big|+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$
→
Find the equation of the plane through (2, 3, -4) and (1, -1, 3) and parallel to x-axis.
→
Find the equation of the plane which contains two parallel lines $\frac{\text{x}-4}{1}=\frac{\text{y}-3}{-4}=\frac{\text{z}-2}{5}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}+2}{-4}=\frac{\text{z}}{5}.$
→
In the set Z of all integers, which of the following relation R is not an equivalence relation?
  1. xRy : if $\text{x}\leq\text{y}$
  2. xRy : if x = y
  3. xRy : if x - y is an even integer
  4. xRy : if $\text{x}\equiv\text{y}\ (\text{mod 3})$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{1}\sqrt{\text{x}(1-\text{x})}\text{ dx}$
→
If $\text{y}\sqrt{\text{x}^2+1}=\log\Big(\sqrt{\text{x}^2+1}-\text{x}\Big),$ prove that $\big(\text{x}^2+1\big)\frac{\text{dx}}{\text{dx}}+\text{xy}+1=0$
→
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?
→
Water is running into an inverted cone at the rate of $\pi$ cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5m. How fast the water level is rising when the water stands 7.5m below the base.
→
Find the area of the region common to the parabolas $4y^2= 9x$ and $3x^2= 16y$.
→
Evaluate the following integrals as limit of sum:
$\int\limits^2_{0}\text{e}^{\text{x}}\text{ dx}$
→