Evaluate the following integrals: x 3x^4-18x^2+11 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}}{3\text{x}^4-18\text{x}^2+11}\text{dx}$

✓

Answer

$\int\frac{\text{x dx}}{3\text{x}^4-18\text{x}^2+11}$
Let $\text{x}^2=\text{t}$
$\Rightarrow2\text{x}\text{ dx = dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
Now, $\int\frac{\text{x dx}}{3\text{x}^4-18\text{x}^2+11}$
$=\frac{1}2{}\int\frac{\text{dt}}{3\text{t}^2-18\text{t}+11}$
$=\frac{1}{3\times2}\int\frac{\text{dt}}{\text{t}^2-6\text{t}+\frac{11}{3}}$
$=\frac{1}{6}\int\frac{\text{dt}}{\text{t}^2-6\text{t}+9-9+\frac{11}{3}}$
$=\frac{1}{6}\int\frac{\text{dt}}{(\text{t}-3)^2-\frac{16}{3}}$
$=\frac{1}{6}\int\frac{\text{dt}}{(\text{t}-3)^2-\Big(\frac{4}{\sqrt{3}}\Big)^2}$
$=\frac{1}{6}\times\frac{1}{2\times\frac{4}{\sqrt{3}}}\log\Bigg|\frac{\text{t}-3-\frac{4}{\sqrt{3}}}{\text{t}-3+\frac{4}{\sqrt{3}}}\Bigg|+\text{C}$
$=\frac{\sqrt{3}}{48}\log\Bigg|\frac{\text{x}^2-3-\frac{4}{\sqrt{3}}}{\text{x}^2-3+\frac{4}{\sqrt{3}}}\Bigg|+\text{C}$

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 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

Find the general solution of the differential equation
$\text{x}\log\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}=\frac{2}{\text{x}}\cdot\log\text{x}$.

Using properties of determinants, prove the following: $ \begin{bmatrix} \text{ x}&\text{x + y }&\text{x} + 2\text{y}\\ \text{x} + 2\text{y} & \text{x}& \text{x + y }\\\text{x + y}&\text{x} + 2\text{y}& \text{x} \end{bmatrix} = 9\text{y}^{2}(\text{x} + \text{y}). $

A firm manufactures headache pills in two sizes A and B. Size A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine; size B contains 1 grain of aspirin, 8 grains of bicarbonate and 66 grains of codeine. It has been found by users that it requires at least 12 grains of aspirin, 7.4 grains of bicarbonate and 24 grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief. Determine also the quantity of codeine consumed by patient

If $A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]$, prove that $\mathrm{A}^3-6 \mathrm{~A}^2+7 \mathrm{~A}+2 \mathrm{I}=0$.

Find the value of p for when the curves $\text{x}^{2} = \text{9p (9 – y) and x}^{2} = \text{p (y + 1)}$ cut each other at right angles

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^2+\text{x}^2-16\text{x}+20}{(\text{x}-2)^2},&\text{ x}\neq2\\\text{k},&\text{x}=2\end{cases}$

Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.

Solve the following differential equation
$\sqrt{1-\text{x}^4}\text{dy}=\text{x dx}$

A company sells two different products, $A$ and $B$. The two products are produced in a common production process, which has a total capacity of $500$ man-hours. It takes 5 hours to produce a unit of A and $3$ hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is $70$ and that for B is $125$. If the profit is Rs. $20$ per unit for the product A and Rs. $15$ per unit for the product B, how many units of each product should be sold to maximize profit?

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.