Evaluate the following definite integrals : _1^2 3 x 9 x^2-1 d x - Vidyadip

Question

Evaluate the following definite integrals : $\int_1^2 \frac{3 x}{9 x^2-1} d x$

✓

Answer

Let $I =\int_1^2 \frac{3 x}{9 x^2-1} d x=\int_1^2 \frac{3 x}{(3 x)^2-1} d x$
Put $3 x = t$
$\therefore 3 dx = dt$
$\therefore dx =\frac{d t}{3}$
When $x=1, t=3 \times 1=3$
When $x=2, t=3 \times 2=6$
$
\begin{aligned}
\therefore I & =\int_3^6 \frac{t}{t^2-1} \cdot \frac{d t}{3}=\frac{1}{6} \int_3^6 \frac{2 t}{t^2-1} d t \\
& =\frac{1}{6}\left[\log \left|t^2-1\right|\right]_3^6 \quad \ldots\left[\because \frac{d}{d t}\left(t^2-1\right)=2 t\right] \\
& =\frac{1}{6}[\log 35-\log 8] \\
& =\frac{1}{6} \log \left(\frac{35}{8}\right) .
\end{aligned}
$
Alternative Method :
$
\begin{aligned}
& \int_1^2 \frac{3 x}{9 x^2-1} d x \\
& =\frac{1}{6} \int_1^2 \frac{18 x}{9 x^2-1} d x \\
& =\frac{1}{6}\left[\log \left|9 x^2-1\right|\right]_1^2 \\
& =\frac{1}{6}[\log 35-\log 8] \\
& =\frac{1}{6} \log \left(\frac{35}{8}\right) .
\end{aligned}
$

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 "Solve the Following Question.(2 Marks)" from Definite Integration (p-1)→Definite Integration (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Obtain the differential equation by eliminating arbitrary constants from the following equations : $y^2=(x+c)^3$

Evalute : $\int \frac{1}{a^2-b^2 x^2} d x$

If $y=[\log (\log (\log x))]^2$, find $\frac{d y}{d x}$

Find $\frac{d y}{d x}$ if, :
$
x=e^{3 t}, y=e^{(4 t+5)}
$

Evalute : $\int \frac{d x}{5-16 x^2}$

Form the differential equation from the relation $x^2+4 y^2=4 b^2$.

Following is the probability distribution of an r.v. X.

X	-3	-2	-1	0	1	2	3
P(X=x)	0.05	0.1	0.15	0.20	0.25	0.15	0.1

Find the probability that
(i) X is positive.
(ii) X is non-negative.
(iii) X is odd.
(iv) X is even.

Represent the following statements by Venn diagrams : If n is a prime number and n ≠ 2, then it is odd.

If $X$ has Poisson distribution with parameter m and $P(X = 2) = P(X = 3)$, then find $P(X ≥ 2)$. Use $e^{-3} = 0.0497$

Construct the truth table for each of the following statement patterns : (p ∨ r) → ~(q ∧ r)