Evaluate the following intregals: 1 x[6( )^2+7 +2] dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{dx}}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}$ $=\int\frac{1}{\text{x}(2\log\text{x}+1)(3\log\text{x}+2)}\text{ dx}$ Now,Let $\frac{1}{\text{x}(2\log\text{x}+1)(3\log\text{x}+2)}=\frac{\text{A}}{\text{x}(2\log\text{x}+1)}+\frac{\text{B}}{\text{x}(3\log\text{x}+2)}$
$\Rightarrow1=\text{A}(3\log\text{x}+2)+\text{B}(2\log\text{x}+1)$ Put $\text{x}=10^{-\frac12{}{}}$ $\Rightarrow1=\frac{1}{2}\text{A}\Rightarrow\text{A}=2$ $-\frac{2}{3}$ Put $\text{x}=10)^{-\frac23}$ $\Rightarrow1=-\frac{1}{3}\text{B}\Rightarrow\text{B}=-3$ $\therefore\text{I}=\int\frac{2\text{dx}}{\text{x}(2\log\text{x}+1)}-\int\frac{3\text{dx}}{\text{x}(3\log\text{x}+2)}$ $=\log|2\log\text{x}+1|-\log|3\log\text{x}+2|\text{C}$ $\therefore\text{I}=\log\Big|\frac{2\log\text{x}+1}{3\log\text{x}+2}\Big|+\text{C}$

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

If $\text{A}=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix},$ show that $A^2 - 7A + 10I_3 = 0.$

Compute the adjoint of the following matrices:$\begin{bmatrix}2 & 0 & -1 \\5 & 1 & 0 \\ 1 & 1 & 3 \end{bmatrix}$
Verify that (adj A)A = |A| I = A (adj A) for the above matrices.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{ax}\text{ at }\Big(\frac{\text{a}}{\text{m}^2},\frac{2\text{a}}{\text{m}}\Big)$

If $\text{y}=\text{x}\sin(\text{a}+\text{y}),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin(\text{a}+\text{y})-\text{y}\cos(\text{a}+\text{y})}$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\frac{2\text{x}+\text{x}^2}{\text{x}}, & \text{x} \neq0\\0,&\text{ x} = 0\end{cases}\text{at x}=0$

Solve the following differential equation:
$(\text{x + y})(\text{dx}-\text{dy})=\text{dx + dy}$

$\int_0^\pi x \cdot \sin x \cdot \cos ^2 x \cdot d x$

A and B throw a pair of dice alternately. A wins the game if he gets a total of $7$ and B wins the game if he gets a total of $10$. If A starts the game, then find the probability that B wins.

Find the nth derivative of the following:

$\frac{1}{x}$

If A and B are two events such that,
$\text{P(A)}=\frac{1}{2},\text{P(B)}=\frac{1}{3}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{4},$ then find $\text{P}(\text{A}|\text{B}), \text{ P}(\text{B}|\text{A}), \text{ P}(\overline{\text{A}}|\text{B})$ and $\text{P}(\overline{\text{A}}|\overline{\text{B}}).$