Integrate the function 1 1 + 4 x^2 - Vidyadip

Question

Integrate the function $\frac{1}{{\sqrt {1 + 4{x^2}} }}$

✓

Answer

$\int {\frac{1}{{\sqrt {1 + 4{x^2}} }}} dx$ $= \int {\frac{1}{{\sqrt {{{\left( {2x} \right)}^2} + {{\left( 1 \right)}^2}} }}} dx$
$= \frac{{\log \left| {\left( {2x} \right) + \sqrt {{{\left( {2x} \right)}^2} + {1^2}} } \right|}}{2}+c$ ...[dividing by 2 as coefficient of x is 2]
$\left[ {\because \int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = \log \left| {x + \sqrt {{x^2} + {a^2}} } \right|} } \right]$
$= \frac{1}{2}\log \left| {2x + \sqrt {4{x^2} + 1} } \right| + c$

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 "3 Marks Question" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the value of $\sin^{-1}(\sin(-600^\circ))\sin(-600^\circ).$

Find the mean and standard deviation of the following probability distributions$:$

$x_i$	$2$	$3$	$4$
$p_i$	$0.2$	$0.5$	$0.3$

Find the principal value of the following:
$\sec^{-1}\Big(2\tan\frac{3\pi}{4}\Big)$

Reduce the equation of the following planes to intercept from and find the intercepts on the coordinate axes:
4x + 3y - 6z - 12 = 0

Write a value of $\int\frac{1+\cot\text{x}}{\text{x}+\log\sin\text{x}}\text{ dx}$

Find the mean and standard deviation of the following probability distributions:

$\text{x}_\text{i}$	$0$	$1$	$2$	$3$	$4$	$5$
$\text{p}_\text{i}$	$\frac{1}{6}$	$\frac{5}{18}$	$\frac{2}{9}$	$\frac{1}{6}$	$\frac{1}{9}$	$\frac{1}{18}$

Evaluate the following integrals:$\int\frac{\log(\log\text{x})}{\text{x}}\text{dx}$

Find a vector of magnitude 5 units and parallel to the resultant of the vectors $\vec a = 2\hat i + 3\hat j - \hat k$ and $\vec b = \hat i - 2\hat j + \hat k$

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)=5\ \text{and}\ \vec{\text{r}}.\Big(3\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)=6.$

State with reasons whether the following functions have inverse:
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}