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

Question

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

✓

Answer

Let $\int\frac{\text{x}^2+1}{(\text{x}-2)^2(\text{x}+3)}=\frac{\text{A}}{\text{x}-2}+\frac{\text{B}}{(\text{x}-2)^2}+\frac{\text{C}}{\text{x}+3}$

$\Rightarrow\text{x}^2+1=\text{A}(\text{x}-2)(\text{x}+3)(\text{x}+3)+\text{B}(\text{x}+3)+\text{C}(\text{x}-2)^2$

$=(\text{A}+\text{C})\text{x}^2+(\text{A}+\text{B}-4\text{C})\text{x}+(-6\text{A}+3\text{B}+4\text{C})$

Equating similar terms, we get,

A + C = 1, A + B - 4C = 0, -6A + 3B + 4C = 1

Solving we get, $\text{A}=\frac{3}{5},\text{B}=1,\text{C}=\frac{2}{5}$

Thus,

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

$\text{I}=\frac{3}{5}\log\text{x}-2|-\frac{1}{(\text{x}-2)}+\frac{2}{5}\log|\text{x}+3|+\text{C}$

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 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

Solve the following system of equations by matrix method:

x - y + 2z = 7

3x + 4y - 5z = -5

2x - y + 3z = 12

Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).

Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-2\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}$

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

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Using Cofactors of elements of third column, evaluate $\triangle=\begin{vmatrix}1&x&yz\\1&y&zx\\1&z&xy\end{vmatrix}.$

Form the differential equation of the family of curves represented by the equation (a being the perimeter):

$(2\text{x}+\text{a})^2+\text{y}^2=\text{a}^2$

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes.

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Solve the following initial value problems:
$\text{xe}^{\frac{\text{y}}{\text{x}}}-\text{y + x}\frac{\text{dy}}{\text{dx}}=0,\text{y(e)}=0$