Prove the following Exercise: ^ 3 _ 1 dx x^ 2 (x+1) =2 3 + 2 3 - Vidyadip

Question

Prove the following Exercise:
$\int^{3}\limits_{1}\frac{\text{dx}}{\text{x}^{2}(\text{x}+1)}=\frac{2}{3}+\log\frac{2}{3}$

✓

Answer

$\text{Let I}=\int^{3}\limits_{1}\frac{\text{dx}}{\text{x}^{2}(\text{x}+1)}$
Also, $\text{Let I}\int^{3}\limits_{1}\frac{1}{\text{x}^{2}(\text{x}+1)}=\frac{\text{A}}{\text{x}}+\frac{\text{B}}{\text{x}^{2}}+\frac{\text{C}}{\text{x}+1}$
$\Rightarrow1=\text{Ax(x}+1)+\text{B}\text{(x}+1)+\text{C}(\text{x}^{2})$
$\Rightarrow1=\text{Ax}^{2}+\text{Ax}+\text{Bx}+\text{B}+\text{Cx}^{2}$
Equating the Coefficient of $\text{x}^{2},\text{x}$ and constant term, we obtain
$\text{A}+\text{C}=0$
$\text{A}+\text{B}=0$
$\text{B}=1$
On solving these equation, we obtain
$\text{A}=-1,\text{C}=1,\ \text{and B}=1$
$\therefore\frac{1}{\text{x}^{2}(\text{x}+1)}=-\frac{1}{\text{x}}+\frac{1}{\text{x}^{2}}+\frac{1}{(\text{x}+1)}$
$\Rightarrow\text{I}=\int^{3}\limits_{1}\left\{-\frac{1}{\text{x}}+\frac{1}{\text{x}^{2}}+\frac{1}{\text{(x}+1)}\right\}\text{dx}$
$=\bigg[-\log\text{x}-\frac{1}{\text{x}}+\log(\text{x}+1)\bigg]^{3}_{1}$
$=\bigg[-\log\bigg(\frac{\text{x}+1}{\text{x}}\bigg)-\frac{1}{\text{x}}\bigg]^{3}_{1}$
$=\log4-\log3-\log2+\frac{2}{3}$
$=-\log2-\log3+\frac{2}{3}$
$=\log\Big(\frac{2}{3}\Big)+\frac{2}{3}$
Hence, the given result is proved

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 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

If $\text{x}=\cos\theta,\text{y}=\sin^3$ prove that $\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}^2}\Big)=3\sin^2\theta(5\cos^2\theta-1)$

A firm manufactures two products A and B. Each product is processed on two machines M₁ and M₂. Product A requires 4 minutes of processing time on M₁ and 8 min. on M₂; product B requires 4 minutes on M₁ and 4 min. on M₂. The machine M₁ is available for not more than 8 hrs 20 min. while machine M₂is available for 10 hrs. during any working day. The products A and B are sold at a profit of Rs. 3 and Rs. 4 respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Evaluate the following intregals:
$\int\frac{\text{x}+1}{\sqrt{4+5\text{x}-\text{x}}}\text{ dx}$

Form the differential equation corresponding to $(\text{x}-\text{a})^2+(\text{y}-\text{b})^2=\text{r}^2$ by eliminating a and b.

Solve the following system of equations by matrix method:

x - y + 2z = 7

3x + 4y - 5z = -5

2x - y + 3z = 12

Evaluate the following integrals:

$\int\sin^{-1}\sqrt{\text{x}}\text{dx}$

A function f : R → R satisfies the equation f(x + y) = f(x)f(y) fot all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f'(0) = 2. Prove that f'(x) = 2 f(x).

Maximize Z = 50x + 30y
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$

Evaluate $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \sin ^2 x d x$.

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