Evaluate the following definite integrals: _ 0 ^21 3+2x-x^2 dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{3+2\text{x}-\text{x}^2}}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{-\text{x}^2+2\text{x}-1+1+3}}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{2}\frac{1}{\sqrt{-(\text{x}-1)^2+4}}\text{ dx}$
$\Rightarrow\text{I}=\Big[\sin^{-1}\frac{(\text{x}-1)}{2}\Big]^{2}_0$
$\Rightarrow\text{I}=\sin^{-1}\frac{1}{2}-\sin^{-1}\Big(-\frac{1}{2}\Big)$
$\Rightarrow\text{I}=2\sin^{-1}\frac{1}{2}$
$\Rightarrow\text{I}=2\times\frac{\pi}{6}$
$\Rightarrow\text{I}=\frac{\pi}{3}$

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 Definite Integrals→Definite 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 f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog.

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$

Evaluate the following integrals:
$\int\frac{\log\text{x}^2}{\text{x}}\text{dx}$

Integrate the following functions w. r. t. x

$\int \frac{1}{3+2 \sin x} \cdot d x$

Evaluate the following integrals:$\int\frac{\text{x}}{\sqrt{4-\text{x}^4}}\text{ dx}$

The probability that certain kind of component will survive a check test is $0.6.$ Find the probability that exactly $2$ of the next $4$ tested components survive

Differentiate the following functions with respect to x:
$\text{e}^{\sin^{-1}2\text{x}}$

Evaluate : $\int e^{\tan ^{-1} x} \cdot\left(\frac{1+x+x^2}{1+x^2}\right) \cdot d x$

Show that the homogeneous equation of degree $2$ in $x$ and $y$ represents a pair of lines passing through the origin if $h^2− ab \geq 0.$

A die is thrown three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ and $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$, if
A = 4 appears on the third toss,
B = 6 and 5 appear respectively on first two tosses.