2x-1 (x-1)^2 dx - Vidyadip

Question

$\int\frac{2\text{x}-1}{(\text{x}-1)^2}\text{ dx}$

✓

Answer

$\int\Big[\frac{2\text{x}-1}{(\text{x}-1)^2}\text{ dx}\Big]$
Let $\text{x}-1=\text{t}$
$\Rightarrow\text{x}=1+\text{t}$
$\Rightarrow1=\frac{\text{dt}}{\text{dx}}$
Now, $\int\Big[\frac{2\text{x}-1}{(\text{x}-1)^2}\text{ dx}\Big]$
$=\int\Big[\frac{2(\text{t}+1)-\text{t}}{\text{t}^2}\Big]\text{dt}$
$=\int\Big(\frac{2\text{t}+1}{\text{t}^2}\Big)\text{dt}$
$=2\int\frac{\text{dt}}{\text{t}}+\int\text{t}^{-2}\text{dt}$
$=2\log|\text{t}|+\frac{\text{t}^{-2+1}}{-2+1}+\text{C}$
$=2\log(\text{x}-1)-\frac{1}{\text{x}-1}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\lim\limits_{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}$ exists finitely, write the value of $\lim\limits_{\text{x}\rightarrow{\text{c}}}\text{f(x)}.$

Show that the line $\frac{\text{x}+3}{-3}=\frac{\text{y}-1}{1}=\frac{\text{z}-5}{5}$ and $\frac{\text{x}+1}{-1}=\frac{\text{y}-2}{2}=\frac{\text{z}-5}{5}$ are coplanar. Hence, find the equation of the plane containing these lines.

Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}-2\text{xy}=(\text{x}^2+2)(\text{x}^2+1)$

Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}-1}{(\text{x}+1)^2(\text{x}+2)}\ \text{dx}$

Show that $\frac{9 \pi}{8}-\frac{9}{4} \sin ^{-1} \frac{1}{3}=\frac{9}{4} \sin ^{-1} \frac{2 \sqrt{2}}{3}$.

Question is modified

Show that $\frac{9 \pi}{8}-\frac{9}{4} \sin ^{-1}\left(\frac{1}{3}\right)=\frac{9}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$.

Let X denote the sum of numbers obtained when two fair dice are rolled. Find the standard deviation of X.

The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?

Evaluate the following integrals:
$\int\frac{\cos^5\text{x}}{\sin\text{x}}\text{ dx}$

Test whether the following relations $R_3$ are:

Reflexive.
Symmetric.
Transitive.

$R_3$ on R defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$

Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\text{xy}=\text{c}^2$