Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following intregals:
$\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$
✓

Answer

 We have
$\text{I}=\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$
Let $\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}}{\text{x}+2}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+2)}{(\text{x}+2)(\text{x}^2+1)}$
$\Rightarrow5=\text{A}(\text{x}^2+1)+\text{Bx}^2+2\text{Bx}+\text{Cx}+2\text{C}$
$\Rightarrow5=(\text{A}+\text{B})\text{x}^2+(2\text{B}+\text{C})\text{x}+(\text{A}+2\text{C})$
Equating coefficient of like terms
A + B = 0 ...(1)
2B + C = 0 ...(2)
A + 2C = 5 ...(3)
Solving (1), (2) and (3), we get
A = 1
B = -1
C = 2
$\therefore\frac{5}{(\text{x}+2)(\text{x}^2+1)}=\frac{1}{\text{x}+2}+\Big(\frac{-\text{x}+2}{\text{x}^2+1}\Big)$
$\Rightarrow\int\frac{5\text{dx}}{(\text{x}+2)(\text{x}^2+1)}+\int\frac{\text{dx}}{\text{x}+2}-\int\frac{\text{x dx}}{\text{x}^2+1}+2\int\frac{\text{dx}}{\text{x}^2+1}$
Let $\text{x}^2+1=\text{t}$
$\Rightarrow2\text{xdx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\int\frac{\text{dx}}{\text{x}+2}-\frac{1}{2}\int\frac{\text{dt}}{\text{t}}+2\int\frac{\text{dx}}{\text{x}^2+1^2}$
$=\log|\text{x}+2|-\frac12\log|\text{t}|+2\tan^{-1}\text{x}+\text{C}$
$=\log|\text{x}+2|-\frac{1}{2}\log|\text{x}^2+2|+2\tan^{-1}\text{x}+\text{C}$

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 "4 Marks" 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

Show that the following systems of linear equations is inconsistent:
3x - y + 2z = 6,
2x - y + z = 2,
3x + 6y + 5z = 20
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_0\frac{1}{\text{x}+\sqrt{\text{a}^2-\text{x}^2}}\text{ dx}$
→
Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}^2+1)\sqrt{\text{x}}}\text{ dx}$
→
A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2m and volume is 8m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?
→
If $\text{x}\sqrt{1+\text{y}}+\text{y}\sqrt{1+\text{x}}=0,$ prove that $(1+\text{x})^2\frac{\text{dx}}{\text{dx}}+1=0$
→
Show that the following system of linear equations is consistent and also find solution:
2x + 3y = 5
6x + 9y = 15
→
$\int\frac{2-3\text{x}}{\sqrt{1+3\text{x}}}\text{dx}$
→
If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is  $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.
→
Using elementary transformation, find the inverse of each of the matrices, $\begin{bmatrix}2& 0&-1 \\5 & 1&0\\0&1&3 \end{bmatrix}$
→
Solve the system of linear equation, using matrix method 2x + y + z = 1; $x - 2y - z = \frac{3}{2};\,\,3y - 5z = 9$
→