Evaluate 1 3 x^2+8 d x - Vidyadip

Question

Evaluate $\int \frac{1}{\sqrt{3 x^2+8}} d x$

✓

Answer

$\text { Let } I =\int \frac{1}{\sqrt{3 x^2+8}} d x$
$=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^2+\frac{8}{3}}} d x$
$=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^2+\left(\frac{2 \sqrt{2}}{\sqrt{3}}\right)^2}} d x$
$=\frac{1}{\sqrt{3}} \log \left|x+\sqrt{x^2+\left(\frac{2 \sqrt{2}}{\sqrt{3}}\right)^2}\right|+ c _1$
$=\frac{1}{\sqrt{3}} \log \left|x+\sqrt{x^2+\frac{8}{3}}\right|+ c _1$
$=\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{3} x+\sqrt{3 x^2+8}}{\sqrt{3}}\right|+ c _1$
$=\frac{1}{\sqrt{3}} \log \left|\sqrt{3} x+\sqrt{3 x^2+8}\right|-\frac{1}{\sqrt{3}} \log \sqrt{3}+ c _1$
$\therefore I =\frac{1}{\sqrt{3}} \log \left|\sqrt{3} x+\sqrt{3 x^2+8}\right|+ c ,$
$\text { where } c = c _1-\frac{1}{\sqrt{3}} \log \sqrt{3}$

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 Question Bank [2022]→Question Bank [2022] — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Assuming that the following statements are true:
p : Sunday is a holiday.
q : Ram does not study on holiday.
Find the truth values of the following statements:
(i) Sunday is not holiday or Ram studies on holiday.
(ii) If Sunday is not a holiday, then Ram studies on holiday.
(iii) Sunday is a holiday and Ram studies on holiday.

A bill of ₹ $65,700$ drawn on July $10$ for $6$ months was discounted for ₹ $65,160$ at $5\%$ p.a. on what day was the bill discounted?

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular
1. $\left[\begin{array}{ccc}2 & 0 & 0 \\ 3 & -1 & 0 \\ -7 & 3 & 1\end{array}\right]$
2. $\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right]$
3. $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

If $A=\left[a_{i j}\right]_{3 \times 3}$ where $a_{i j}=2(i-j)$. Find $A$ and $A^{\top}$. State whether $A$ and $A^{\top}$ both are symmetric or skew-symmetric matrices.

Find the area of the region bounded by the curve $y=\sqrt{2 x+3}$, the $X$ axis and the lines $x=0$ and $x=2$

Find the area of the region bounded by the curve $y=\sqrt{36-x^2}$, the $X$-axis lying in the first quadrant and the lines $x=0$ and $x=6$

Solve the following differential equations : $\left(x^2-y x^2\right) d y+\left(y^2+x y^2\right) d x=0$

Find MPC, MPS, APC and APS, if the expenditure $E_c$ of a person with income $I$ is given as $E_C=\left.(0.0003)\right|^2+(0.075) I$, when $I=1000$.

Ms. Mohana want to invest at least ₹ 55000 in Mutual funds and fixed deposits. Mathematically this information can be written as ______

Obtain trend values for data in Problem 16 using 3-yearly moving averages.