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

Question

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

✓

Answer

Let $\text{I}=\int\frac{3\text{x}-2}{(\text{x}+1)^2(\text{x}+3)}\ \text{dx}$
We express
$\frac{3\text{x}-2}{(\text{x}+1)^2(\text{x}+3)}=\frac{\text{A}}{\text{x}+1}+\frac{\text{B}}{(\text{x}+1)^2}+\frac{\text{C}}{\text{x}+3}$
$\Rightarrow3\text{x}-2=\text{A}(\text{x}+1)(\text{x}+3)+\text{B}(\text{x}+3)+\text{C}(\text{x}+1)^2$
Equating the coefficient of $x^2, x$ and constants, we get
$0 = A + C$ and $3 = 4A + B + 2C$ and $-2 = 3A + 3B + C$
$\text{or }\text{A}=\frac{11}{4}\text{ and }\text{B}=-\frac{5}{2}\text{ and }\text{C}=-\frac{11}{4}$
$\therefore\text{I}=\int\bigg(\frac{\frac{11}{4}}{\text{x}+1}+\frac{-\frac{5}{2}}{(\text{x}+1)^2}+\frac{-\frac{11}{4}}{\text{x}+3}\bigg)\ \text{dx}$
$=\frac{11}{4}\int\frac{1}{\text{x}+1}\text{ dx }-\frac{5}{2}\int\frac{1}{(\text{x}+1)^2}\text{ dx }-\frac{11}{4}\int\frac{1}{\text{x}+3}\text{ dx}$
$=\frac{11}{4}\log|\text{x}+1|+\frac{5}{2(\text{x}+1)}-\frac{11}{4}\log|\text{x}+3|+\text{C}$
Hence,
$\int\frac{3\text{x}-2}{(\text{x}+1)^2(\text{x}+3)}\ \text{dx}=\frac{11}{4}\log|\text{x}+1|+\frac{5}{2(\text{x}+1)}-\frac{11}{4}\log|\text{x}+3|+\text{C}$

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 "Solve the Following Question.(3 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

Show that the points $A(2,-1,0) B(-3,0,4), C(-1,-1,4)$ and $D(0,-5,2)$ are non coplanar.

Evaluate : $\int_1^3 \frac{1}{\sqrt{2+x}+\sqrt{x}} \cdot d x$

A function f(x) is defined as,
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-\text{x}-6}{\text{x}-3}&; &\text{if} \text{x}\neq3\\5 &;&\text{if}\text{ x}=3\end{cases}$
show that f(x) is continuous that x = 3.

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers. Find E(X).

Show that the four points having position vectors $6\hat{\text{i}}-7\hat{\text{j}},\ 16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},\ 3\hat{\text{j}}-6\hat{\text{k}},\ 2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

Let * be a binary operation on $Q_0$ (set of non-zero rational numbers) defined by $\text{a}\ ^* \ \text{b}=\frac{\text{ab}}{5}$ for all $\text{a, b}\in\text{Q}_0.$ Show that * is commutative as well as associative. Also, find its identity element if it exists.

Find the magnitude of $\vec{\text{a}}=\big(3\hat{\text{k}}+4\hat{\text{j}}\big)\times(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}).$

Three machines $E_1, E_2, E_3$ in a certain factory produce $50\%, 25\%$ and $25\%$, respectively, of the total daily output of electric bulbs. It is known that $4\%$ of the tubes produced one each of the machines $E_1$ and $E_2$ are defective, and that $5\%$ of those produced on $E_3$ are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Write the value of $\cos^{-1}(\cos350^\circ)-\sin^{-1}(\sin350^\circ)$