Evaluate the following integrals: 1 7-3x-2x^2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{\sqrt{7-3\text{x}-2\text{x}^2}}\text{ dx}$

✓

Answer

$\int\frac{\text{dx}}{\sqrt{7-3\text{x}-2\text{x}^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}-\frac{3}{2}\text{x}-\text{x}^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}-\big(\text{x}^2-\frac{3}{2}\text{x}\big)}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt7}{\sqrt2}\Big)^2-\Big(\text{x}^2+\frac{3}{2}\text{x}+\big(\frac{3}{4}\big)^2-\big(\frac{3}{4}\big)^2\Big)}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt7}{\sqrt2}\Big)^2-\big(\text{x}+\frac{3}{4}\big)^2+\frac{9}{16}}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{7}{2}+\frac{9}{16}-\big(\text{x}+\frac{3}{4}\big)^2}}$ $=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\frac{56+9}{16}-\big(\text{x}+\frac{3}{4}\big)^2}}$$=\frac{1}{\sqrt2}\int\frac{\text{dx}}{\sqrt{\Big(\frac{\sqrt{65}}{4}\Big)^2-\big(\text{x}+\frac{3}{4}\big)^2}}$
$=\frac{1}{\sqrt2}\sin^{-1}\Bigg[\frac{\text{x}+\frac{3}{4}}{\frac{\sqrt{65}}{4}}{}\Bigg]+\text{C}$ $=\frac{1}{\sqrt2}\sin^{-1}\Big[\frac{4\text{x}+3}{\sqrt{65}}\Big]+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Prove that:
$\begin{vmatrix}\text{a}+\text{b}&\text{b}+\text{c}&\text{c}+\text{a}\\\text{b}+\text{c}&\text{c}+\text{a}&\text{a}+\text{b}\\\text{c}+\text{a}&\text{a}+\text{b}&\text{b}+\text{c}\end{vmatrix}=2\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{b}&\text{c}&\text{a}\\\text{c}&\text{a}&\text{b} \end{vmatrix}$

Find the shortest distance between the following pairs of lines whose cartesian equation are:
$\frac{\text{x}-1}{-1}=\frac{\text{y}+2}{1}=\frac{\text{z}-3}{-2}$ and $\frac{\text{x}-1}{1}=\frac{\text{y}+1}{2}=\frac{\text{z}+1}{-2}$

Evaluate the following integrals:
$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$

Consider the binary operation $^*$ and o defined by the following tables on set $S = {a, b, c, d}.$

$o$	$a$	$b$	$c$	$d$
$a$	$a$	$a$	$a$	$a$
$b$	$a$	$b$	$c$	$d$
$c$	$a$	$c$	$d$	$b$
$d$	$a$	$d$	$b$	$c$

Using properties of determinants, prove that:
$\begin{vmatrix}3\text{a}&-\text{a+b}&-\text{a+c}\\-\text{b+a}&3\text{b}&-\text{b+c}\\-\text{c+a}&\text{c+b}&3\text{c}\end{vmatrix}=3 (\text{a + b + c}) (\text{ab + bc + ca})$

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as

number greater than 4
six appears on at least one die

Verify Rolle's theorem for the function $f(x) = x^2 - 4x + 3$ on $[1, 3].$

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Finde the value of a and b, if the function f(x) defined by $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, &\text{x}\leq1\\\text{bx}+2, & \text{x}>1\end{cases}$is differentiable at x = 1.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\text{a}\sin\text{x}+\text{b}\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$