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

Question

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

✓

Answer

$\text{Here I}=\int\frac{\text{x+2}}{\text{x}^2-5\text{x}+6}\text{ dx}=\frac{1}{2}\int\frac{2\text{x}-5+9}{\sqrt{{\text{x}^2}-5\text{x}+6}}\text{dx} $
$=\frac{1}{2}\int\frac{2\text{x}-5}{\sqrt{\text{x}^2-5\text{x}+6}}\ \text{dx}+\frac{9}{2}\int\frac{1}{\sqrt{\bigg(\text{x}-{5}/{2}\bigg)^2-\bigg(\frac{1}{2}\bigg)^2}}\ \text{dx}$
$\sqrt{\text{x}^2+4\text{x}+3}+\frac{9}{2}\log\begin{vmatrix}\bigg(\text{x}-\frac{5}{2}\bigg)+\sqrt{\text{x}^2-5\text{x}+6}\end{vmatrix}+\text{c}$

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 INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\tan^{-1}\frac{2\text{a}\text{b}}{\text{a}^2-\text{b}^2}+\tan^{-1}\frac{2\text{xy}}{\text{x}^2-\text{y}^2}=\tan^{-1}\frac{2\alpha\beta}{\alpha^2-\beta^2},$

where $\alpha=\text{ax}-\text{by}$ and $\beta=\text{ay}+\text{bx}.$

If sin y = x sin (a + y), prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^{2}\text{(a + y)}}{\sin\text{a}}.$

Find the angles which the vector $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\sqrt{2}\hat{\text{k}}$ makes with the coordinate axes.

Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}^2}}\Big)$

Evaluate the following integrals:
$\int\frac{\text{x}}{\text{x}^4+2\text{x}^2+3}\text{dx}$

Find the value of k if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ where
$\text{f}\text{(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}, &\text{ x}\neq\frac{\pi}{2}\\3, &\text{ x}=\frac{\pi}{2}\end{cases}$

Evaluate definite integral $\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x$.

If $\text{x}=\text{e}^{\cos2\text{t}}$ and $\text{y}=\text{e}^{\sin2\text{t}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$

Sketch the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 1. Also find the area of this region.

Solve the following LPP graphically:
Minimise Z = 3x + 9y
subject to the constraints
x + 3y $\leq$ 60
x + y $\geq$ 10
x $\leq$ y
x $\geq$ 0, y $\geq$ 0.