Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{3x}}}{4\text{e}^{6\text{x}}-9}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\text{e}^{3\text{x}}}{4\text{e}^{6\text{x}}-9}\text{dx}$
Let $\text{e}^{3\text{x}}=\text{t}$
$\Rightarrow3\text{e}^{3\text{x}}\text{ dx = dt}$
$\Rightarrow\text{e}^{3\text{x}} \text{dx}=\frac{\text{dt}}{3}$
$\text{I}=\frac{1}3{}\int\frac{\text{dt}}{4\text{t}^2-9}$
$=\frac{1}{12}\int\frac{\text{dt}}{\text{t}^2-\frac{9}{4}}$
$=\frac{1}{12}\int\frac{\text{dt}}{\text{t}^2-\big(\frac{3}{2}\big)^2}$
$=\frac{1}{12}\times\frac{1}{2\big(\frac{3}{2}\big)}\log\Bigg|\frac{\text{t}-\frac{3}{2}}{\text{t}+\frac{3}{2}}\Bigg|+\text{C}$ $\bigg[\text{Since,}\int\frac{1}{\text{x}^2+\text{a}^2}\text{dx}=\frac{1}{2\text{a}}\bigg|\frac{\text{x}-\text{a}}{\text{x+a}}\bigg|+\text{C}\bigg]$
$\text{I}=\frac{1}{36}\log\bigg|\frac{2\text{t}-3}{2\text{t}+3}\bigg|+\text{C}$
$\text{I}=\frac{1}{36}\log\bigg|\frac{2\text{e}^{3\text{x}}-3}{2\text{e}^{3\text{x}}+3}\bigg|+\text{C}$

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 "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

Find the angle between the vectors with direction ratios proportional to 1, -2, 1 and 4, 3, 2.
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\sin\text{x}\cos^4\text{x}\text{ dx}$
→
Evaluate the following intregals:
$\int\frac{\text{x}+2}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}$
→
$\int\text{x}\sqrt{\text{x}+2}\ \text{dx}$
→
Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line $\frac{\text{x}-1}{1}=\frac{2\text{y}+1}{2}=\frac{\text{z}+1}{-1}.$ 
→
Evaluate the following integrals:

$\int\sin\text{x}\log(\cos\text{x})\text{dx}$

→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{x}}{1+6\text{x}^3}\Big)$
→
Evaluate:
$\int\frac{\sin \text{x} - \text{x}\cos \text{x}}{\text{x} ( \text{x}+ \sin \text{x})} \text{dx}$
→
Show that $\text{y}=\text{A}\cos\text{x}+\text{B}\sin\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$
→
Prove that: $\big(\vec{\text{a}}-\vec{\text{b}}\big).\big\{\big(\vec{\text{b}}-\vec{\text{c}}\big)\big\}=0$
→