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

Question

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

✓

Answer

Let $\frac{\text{x}}{(\text{x}-1)^2(\text{x}+2)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{B}}{(\text{x}-1)}+\frac{\text{C}}{\text{(x}+1)^2}$
$\Rightarrow x^2 = A(x + 1)^2+ B(x - 1) (x + 1) + C(x - 1)$
$= (A + B) x^2 + (2A + C) x + (A - B - C)$
Equating similar terms,
$A + B = , 2A + C = 0, A - B - C = 0$
Solving, we get, $\text{A}=\frac{1}{4},\text{B}=\frac{3}{4},\text{C}=-\frac{1}{2}$
Thus,
$\text{I}=\frac{1}{4}\int\frac{\text{dx}}{\text{x}-1}+\frac{3}{4}\int\frac{\text{dx}}{\text{x}+1}-\frac{1}{2}\int\frac{\text{dx}}{(\text{x}+1)}^2$
$=\frac{1}{4}\log|\text{x}-1|+\frac{3}{4}\log|\text{x}+1|+\frac{1}{2(\text{x}+1)}+\text{C}$
$\text{I}=\frac{1}{4}\log|\text{x}-1|+\frac{3}{4}\log|\text{x}+1|+\frac{1}{2(\text{x}+1)}\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

An insurance company insured $3000$ scooters, $4000$ cars and $5000$ trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02, 0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident. Find the probability that it is a,

Scooter.
Car.
Truck.

If O is the origin and the coordinates of A are (a, b, c) Find the direction cosines of OA and the equation of the plane through A at right angles to OA.

Evaluate the following integrals:$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$

Determine the area under the cutve $\text{y}=\sqrt{\text{x}^{2}-\text{x}^{2}}$ included between the lines $x = 0$ and $x = a$.

Find the maximum and minimum of the following functions:

$f(x)=x^2+\frac{16}{x^2}$

Solve the following initial value problems:
$(\text{y}^4-2\text{x}^3\text{y})\text{dx}+(\text{x}^4-2\text{xy}^3)\text{dy}=0,\text{y}(1)=1$

Evaluate the following integrals:$\int_{1}^\limits{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}$

Evaluate the following intregals:
$\int\frac{1}{2+\sin\text{x}+\cos\text{x}}\text{dx}$

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}5,&\text{if }\text{ x}\leq2\\\text{ax}+\text{b},&\text{if }2<\text{x}<10\\21,&\text{if }\text{ x}\geq10\end{cases}$

Show that $\text{f}(\text{x})=\text{x}^\frac{1}{3}$ is not differentible at x = 0.