Evaluate: 3x + 1 2x^ 2 -2x + 3 dx - Vidyadip

Question

Evaluate:

$\int\frac{3x + 1}{2x^{2} -2x + 3} dx$

✓

Answer

$\int\frac{3x + 1}{2x^{2} -2x + 3} = \int \frac{\frac{3}{4}\big(4x -2\big)+\frac{5}{2}}{2x^{2}-2x + 3 }\text{dx}$

$= \frac{3}{4}\int\frac{(4x - 2) dx}{2x^{2} - 2x + 3} + \frac{5}{4} \int \frac{dx}{x^{2} - x +\frac{3}{2}}$

$\frac{3}{4} \log | 2x^{2} - 2x + 3| + \frac{5}{4} \int \frac{dx} {\bigg(x - \frac{1}{2}\bigg)^{2} + \bigg( \frac{\sqrt{5}}{2}\bigg)^{2}} +c$

$= \frac{3}{4} \log | 2x^{2} - 2x + 3| \frac{\sqrt{5}}{2}\tan ^{-1} \frac{2x - 1}{\sqrt{5}}$

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

Differentiate the following functions with respect to x:
$\sqrt{\frac{\text{a}^2-\text{x}^2}{\text{a}^2+\text{x}^2}}$

Prove that the function $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.

Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$

A(1, 0, 4) B(0, -11, 1), C(2, -3, 1) are three points and D is the fool of perpendicular from A on BC. Find the coordinates of D.

Find the equation of the containing the line $\frac{\text{x}+1}{-3}=\frac{\text{y}-3}{2}=\frac{\text{z}+2}{1}$ and the point (0, 7, -7) and show that the line $\frac{\text{x}}{1}=\frac{\text{y}-7}{-3}=\frac{\text{z}+7}{2}$ also lies in the same plane.

Solve the following systems of linear equations by cramer's rule:

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32.

Show that $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$ is decreases in the interval $[0,\infty)$ and increases in the interval $(-\infty,0].$

Evaluate the following integrals:
$\int(3\text{x}+1)\sqrt{4-3\text{x}-2\text{x}^2}\text{dx}$

Find the intervals in which the following functions are increasing or decreasing.
f(x) = x⁴ - 4x³ - 4x² + 15