Question
Find: $\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x$
✓
Answer
We write
$\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C}{x+3}$
So that 3x - 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2
= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1)
Comparing coefficient of x2, x and constant term on both sides,
we get A + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = -2. Solving these equations, we get
$A=\frac{11}{4}, B=\frac{-5}{2}$ and $C=\frac{-11}{4}$. Thus the integrand is given by
$\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{11}{4(x+1)}-\frac{5}{2(x+1)^{2}}-\frac{11}{4(x+3)}$
Therefore, $\int \frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{11}{4} \int \frac{d x}{x+1}-\frac{5}{2} \int \frac{d x}{(x+1)^{2}}-\frac{11}{4} \int \frac{d x}{x+3}$
= $\frac{11}{4} \log |x+1|+\frac{5}{2(x+1)}-\frac{11}{4} \log |x+3|+\mathrm{C}$
= $\frac{11}{4} \log \left|\frac{x+1}{x+3}\right|+\frac{5}{2(x+1)}+C$
$\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C}{x+3}$
So that 3x - 2 = A (x + 1) (x + 3) + B (x + 3) + C (x + 1)2
= A (x2 + 4x + 3) + B (x + 3) + C (x2 + 2x + 1)
Comparing coefficient of x2, x and constant term on both sides,
we get A + C = 0, 4A + B + 2C = 3 and 3A + 3B + C = -2. Solving these equations, we get
$A=\frac{11}{4}, B=\frac{-5}{2}$ and $C=\frac{-11}{4}$. Thus the integrand is given by
$\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{11}{4(x+1)}-\frac{5}{2(x+1)^{2}}-\frac{11}{4(x+3)}$
Therefore, $\int \frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{11}{4} \int \frac{d x}{x+1}-\frac{5}{2} \int \frac{d x}{(x+1)^{2}}-\frac{11}{4} \int \frac{d x}{x+3}$
= $\frac{11}{4} \log |x+1|+\frac{5}{2(x+1)}-\frac{11}{4} \log |x+3|+\mathrm{C}$
= $\frac{11}{4} \log \left|\frac{x+1}{x+3}\right|+\frac{5}{2(x+1)}+C$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
In the given Figure, which vectors are equal?
A relation $R$ is defined as $A R B \Leftrightarrow A$ is subset of $B$ is sets of set S . Is this relation R will anti symmetric?
→Find the magnitude of each of the two vectors $\vec{\text{a}}\ \text{and}\ \vec{\text{b}},$ having the same magnitude such that the angle between them is 60° and their scalar product is $\frac{9}{2}.$
→If the matrix $\text{A}=\begin{bmatrix}0 & \text{a} & -3 \\2 & 0&-1\\\text{b}&1&0 \end{bmatrix}$ is skew symmetric, find the values of ‘a’ and ‘b’.
→Find $\int \sqrt{3-2 x-x^{2}} d x$
→$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$ is equal to
→Show that the matrix $A=\left[\begin{array}{rrr} {0} & {1} & {-1} \\ {-1} & {0} & {1} \\ {1} & {-1} & {0} \end{array}\right]$is a skew-symmetric matrix.
→Prove that the function $f(x) = {x^n}$ is continuous at x = n where n is a positive integer.
→The probability distribution of random variable X is given below:
→| $\text{X}$ | $0$ | $1$ | $2$ | $3$ |
| $\text{P}(\text{X})$ | $\text{k}$ | $\frac{\text{k}}{2}$ | $\frac{\text{k}}{4}$ | $\frac{\text{k}}{8}$ |
Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}>2)$