Evaluate the following intregals: 2x+1 (x-2)(x-3) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{2\text{x}+1}{(\text{x}-2)(\text{x}-3)}\ \text{dx}$

✓

Answer

Let $\frac{2\text{x}+1}{(\text{x}-2)(\text{x}-3)}=\frac{\text{A}}{(\text{x}-2)}+\frac{\text{B}}{(\text{x}-3)}$
$\Rightarrow2\text{x}+1=\text{A}(\text{x}-3)+\text{B}(\text{x}-2)$
$=(\text{A}+\text{B})\text{x}+(-3\text{A}-2\text{B})$
equating similar terms, we get
A + B = 2, and -3A - 2B = 1
Thus,
$\text{I}=-5\int\frac{\text{dx}}{\text{x}-2}+7\int\frac{\text{dx}}{\text{x}-3}$
$=-5\log|\text{x}-2|+7\log|\text{x}-3|+\text{C}$
$\text{I}=\log\Big|\frac{(\text{x}-3)^7}{(\text{x}-2)^5}\Big|+\text{C}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that each of the given three vectors is a unit vector:
$ \frac{1}{7}\left( {2\hat i + 3\hat j + 6\hat k} \right), \frac{1}{7}\left( {6\hat i + 2\hat j - 3\hat k} \right), \ \frac{1}{7}\left( {3\hat i - 6\hat j + 2\hat k} \right)$
Also, show that they are mutually perpendicular to each other.

Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines: $\frac{\text{x}-8}{3}=\frac{\text{y}+19}{-16}=\frac{\text{z}-10}{7}\ \text{and}\ \frac{\text{x}-15}{3}=\frac{\text{y}-29}{8}=\frac{\text{z}-5}{-5}.$

$\text{If y = }\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}},\text{ show that }\text{2x}\frac{\text{dy}}{\text{dx}}+\text{y}=2\sqrt{\text{x}}.$

Find the image of the point with position vector $3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$ in the plane $\vec{\text{r}}. (2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=4.$ Also, find the position vectors of the foot of the prependicular and the equation of the perpendicular line through $3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}.$

Evaluate: $\int\text{x sin}^{-1}\text{x dx}.$

Find the equation of a plane passing through the points $(0, 0, 0)$ and $(3, -1,2)$ and parallel to the line $\frac{\text{x}-4}{1}=\frac{\text{y}+3}{-4}=\frac{\text{z}+1}{7}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}-3|,&\text{if }\text{ x}\geq1\\\frac{\text{x}^2}{4}-\frac{3\text{x}}{2}+\frac{13}{4},&\text{if }\text{ x}<1\end{cases}$

Find the local maximum and local minima, of the function $\text{f(x)} = \sin x - \cos x, 0< x < 2\pi.$ Also find the local maximum and local minimum values.

Show that area of the parallelogram whose diagonals ate given by $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{|\vec{\text{a}}\times\vec{\text{b}}|}{2}.$ Also, find the area of the parallelogram, whose diagonals are $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}.$

Find the general solution $ x \frac { d y } { d x } + y - x + x y \cot x = 0 \ (x \neq 0)$.