Fill in the blanks: If ^a_01 1+4x^2 dx= 8 , then a = ________. - Vidyadip

Question

Fill in the blanks:
If $\int\limits^\text{a}_0\frac{1}{1+4\text{x}^2}\text{dx}=\frac{\pi}{8},$ then a = ________.

✓

Answer

If $\int\limits^\text{a}_0\frac{1}{1+4\text{x}^2}\text{dx}=\frac{\pi}{8},$ then $\text{a}=\frac{1}{2}$
Let $\text{I}=\int\limits^\text{a}_0\frac{1}{1+4\text{x}^2}\text{dx}=\frac{\pi}{8}$
Now, $\int\limits^\text{a}_0\frac{1}{4\Big(\frac{1}{4}+\text{x}^2\Big)}\text{dx}=\frac{2}{4}\big[\tan^{-1}2\text{x}\big]^\text{a}_0$
$=\frac{1}{2}\tan^{-1}2\text{a}-0=\frac{\pi}{8}$
$\frac{1}{2}\tan^{-1}2\text{a}=\frac{\pi}{8}$
$\Rightarrow\ \tan^{-1}2\text{a}=\frac{\pi}{4}$
$\Rightarrow\ 2\text{a}=1$
$\therefore\ \text{a}=\frac{1}{2}$

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 "Fill In The Blanks[1 Marks ]" 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

If the radius of a sphere is changing at the rate of 0.2 cm/sec, then the rate of change in its surface area will be______

For set A , relation R is defined by ________ if every $a \in A,(a, a) \in R$.

The general solution of the differential equation $\frac{d y}{d x}=\frac{1+y^2}{1+x^2}$ will be ___________ .

Fill in the blank.
If $A$ is a skew symmetric matrix, then $A^2$ is a _________.

The order of $y^{\prime \prime \prime}+y^2+e^{y^{\prime}}=0$ will be ___________ .

Fill in the blanks.
The cartesian equation of the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})=2$ is ________.

The equations of a line passing through origin and parallel to $x$-axis are _________ .

Fill in the blanks.
The number of arbitrary constants in the general solution of a differential equation of order three is ________.

The value of $\sin ^{-1}\left(\cos \frac{33 \pi}{5}\right)$ is. ________

Fill in the blanks.
If $\vec{\text{a}}$ is any non-zero vector, then $(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$ equal ________.