Evaluate the following integrals: 1 2x^2-x-1 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{2\text{x}^2-\text{x}-1}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{2\text{x}^2-\text{x}-1}\text{dx}$
$=\frac{1}{2}\int\frac{1}{\text{x}^2-\frac{\text{x}}{2}-\frac{1}{2}}\text{dx}$
$=\frac{1}{2}\int\frac{1}{\text{x}^2-2\text{x}\times\frac{1}{4}+\big(\frac{1}{4}\big)^2-\big(\frac{1}{4}\big)^2-\frac{1}{2}}\text{dx}$
$=\frac{1}{2}\int\frac{1}{\big(\text{x}-\frac{1}{4}\big)^2-\frac{9}{16}}\text{dx}$
Let $\text{x}-\frac{1}{4}=\text{t}$
$\Rightarrow\text{dx = dt}$
$\text{I}=\frac{1}{2}\int\frac{1}{\text{t}^2-\big(\frac{3}{4}\big)^2}\text{dt}$
$\text{I}=\frac{1}{2}\times\frac{1}{2\times\big(\frac{3}{4}\big)}\log\Bigg|\frac{\text{t}-\frac{3}{4}}{\text{t}+\frac{3}{4}}\Bigg|+\text{C} $ $\Big[\text{Since,}\int\frac{1}{\text{x}^2-\text{a}^2}\text{dx}=\frac{1}{2\text{a}}\log\bigg|\frac{\text{x}-\text{a}}{\text{x+a}}\bigg|+\text{C}\Big]$
$\text{I}=\frac{1}{3}\log\Bigg|\frac{\text{x}-\frac{1}{4}-\frac{3}{4}}{\text{x}-\frac{1}{4}+\frac{3}{4}}\Bigg|+\text{C}$
$\text{I}=\frac{1}{3}\log\bigg|\frac{\text{x}-1}{2\text{x}+1}\bigg|+\text{C}$

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

Similar questions

Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$

Compute the adjoint of the following matrices:$\begin{bmatrix}2 & 0 & -1 \\5 & 1 & 0 \\ 1 & 1 & 3 \end{bmatrix}$
Verify that (adj A)A = |A| I = A (adj A) for the above matrices.

Represent the following families of curves by forming the corresponding differential equation:
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1$

Solve the following differential equation:
$\sqrt{\text{1 + x}^{2}+\text{y}^{2}+\text{x}^{2}\text{y}^{2}}+\text{xy}\frac{\text{dy}}{\text{dx}}=0.$

Using differentials, find the approximate values of the following:
$(66)^{\frac{1}{3}}$

Find the angle between the lines whose direction cosines are given by the equations:
$2l + 2m - n = 0, mn + ln + lm = 0$

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Find the maximum and the minimum values, if any, without using derivaives of the following functions:$f(x) = 2x^3+ 5$ on R.

A bag contains $5$ red marbles and $3$ black marbles. Three marbles are drawn one bybone without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?

A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require $4$ sq. metre per box while the small boxes require $3$ sq. metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If $60$ sq. metre of cardboard is in stock, and if the profits on the large and small boxes are Rs. $3$ and Rs. $2$ per box, how many of each should be made in order to maximize the total profit?