Integrate the function in Exercise. 1+ x^2 9 - Vidyadip

Question

Integrate the function in Exercise.
$\sqrt{1+\frac{\text{x}^2}{9}}$

✓

Answer

$\int\sqrt{1+\frac{\text{x}^2}{9}}\text{dx}=\int\sqrt{\frac{9+\text{x}^2}{9}}\text{dx}=\int\frac{\sqrt{\text{x}^2+3^2}}{3}\text{dx}=\frac{1}{3}\int\sqrt{\text{x}^2+3^2}\text{dx}$
$=\frac{1}{3}\Bigg[\frac{\text{x}}{2}\sqrt{\text{x}^2+3^2}+\frac{3}{2}\text{log}\Big|\text{x}+\sqrt{\text{x}^2+3^2}\Big|\Bigg]+\text{c}\Bigg[\therefore\ \int\sqrt{\text{x}^2+\text{a}^2}\text{dx}=\frac{\text{x}}{2}\sqrt{\text{x}^2+\text{a}^2}+\frac{\text{a}^2}{2}\text{log}\Big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\Big|\Bigg]+\text{c}$
$=\frac{\text{x}}{6}\sqrt{\text{x}^2+9}+\frac{3}{2}\text{log}\Big|\text{x}+\sqrt{\text{x}^2+9}\Big|+\text{c}$

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 "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

A function f(x) is defined as,
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-\text{x}-6}{\text{x}-3}&; &\text{if} \text{x}\neq3\\5 &;&\text{if}\text{ x}=3\end{cases}$
show that f(x) is continuous that x = 3.

Probality of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently. Find the probability that the problem is solved.

Let $R^+$ be the set of all non$-$negative real numbers. If $f : R^+ \rightarrow R^+$ and $g : R^+ \rightarrow R^+$ are defined as $f(x) = x^2$ and $\text{g(x)}=+\sqrt{\text{x}},$ find $fog$ and $gof.$ Are they equal functions?

Prove that
$\text{L.H.S}=\sin\Big\{\tan^{-1}\frac{1-\text{x}^2}{2\text{x}}+\cos^{-1}\frac{1-\text{x}^2}{1+\text{x}^2}\Big\}=1$

Evaluate the following integrals:
$\int\frac{1}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}\text{dx}$

A plabne makes intercepts -6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\text{|x|}\cos\Big(\frac{1}{\text{x}}\Big), & \text{ x}\neq 0\\0 &\text{ x} = 0\end{cases}\text{at x}=0$

Evalute the following integrals:
$\int\frac{1}{\text{x}(3+\log\text{x})}\text{dx}$

Find the principal value of the following:
$\cos^{-1}\Big(-\frac{\sqrt3}{2}\Big)$

What is the value of $\cos^{-1}\Big(\cos\frac{2\pi}{3}\Big)+\sin^{-1}\Big(\sin\frac{2\pi}{3}\Big)$