Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
$\int\limits^3_0\frac{3\text{x}+1}{\text{x}^2+9}\text{ dx}=$
  • ✓
    $\frac{\pi}{12}+\log\big(2\sqrt{2}\big)$
  • B
    $\frac{\pi}{2}+\log\big(2\sqrt{2}\big)$
  • C
    $\frac{\pi}{6}+\log\big(2\sqrt{2}\big)$
  • D
    $\frac{\pi}{3}+\log\big(2\sqrt{2}\big)$
✓

Answer

Correct option: A.
$\frac{\pi}{12}+\log\big(2\sqrt{2}\big)$
We have,
$\text{I}=\int\limits^3_0\frac{3\text{x}+1}{\text{x}^2+9}\text{ dx}$
$\text{I}=\int\limits^3_0\frac{3\text{x}}{\text{x}^2+9}\text{ dx}+\int\limits^3_0\frac{1}{\text{x}^2+9}\text{ dx}$
$\text{I}_1=\int\limits^3_0\frac{3\text{x}}{\text{x}^2+9}\text{ dx}$ and $\text{I}_2=\int\limits^3_0\frac{1}{\text{x}^2+9}\text{ dx}$
Putting $\text{x}^2+9=\text{t}$ in  $l_1$     
$\Rightarrow 2\text{x}\text{ dx}=\text{dt}$
$\Rightarrow \text{x}\text{ dx}=\frac{\text{dt}}{2}$
when $\text{x}\rightarrow0;\text{t}\rightarrow9$
and $\text{x}\rightarrow3;\text{t}\rightarrow18$
$\therefore\text{I}=\int\limits^{18}_9\frac{3\text{ dt}}{2\text{ t}}+\int\limits^3_0\frac{1}{\text{x}^2+9}\text{ dx}$
$=\frac{3}{2}\int\limits^{18}_9\frac{\text{dt}}{\text{t}}+\int\limits^3_0\frac{1}{\text{x}^2+3^2}\text{ dx}$
$=\frac{3}{2}\big[\log(\text{t})\big]^{18}_9+\frac{1}{3}\Big[\tan^{-1}\Big(\frac{\pi}{3}\Big)\Big]^3_0$
$=\frac{3}{2}\big[\log18-\log9\big]+\frac{1}{3}\Big(\frac{\pi}{4}-0\Big)$
$=\frac{3}{2}\Big[\log\frac{18}{9}\Big]+\frac{\pi}{12}$
$=\frac{3}{2}\big[\log2\big]+\frac{\pi}{12}$
$=\log(\sqrt{8})+\frac{\pi}{12}$
$=\log(2\sqrt{2})+\frac{\pi}{12}$
$=\frac{\pi}{12}+\log(2\sqrt{2})$

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 "M.C.Q (1 Marks)" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In order to maximize the profit of the company, the optimal solution of which of the following equations is required?
→
Choose the correct answer from the given four option.
Solution of the differential equation $\tan\text{y}\sec^2\text{xdx} + \tan\text{x }\sec^2\text{ydy}=0$is:
→
Let $A$ be a $3 \times 3$ matrix such that $\operatorname{adj} A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right]$ and $B = adj ($ adj $A )$ If $| A |=\lambda$ and $\left|\left( B ^{-1}\right)^{ T }\right|=\mu,$ then the ordered pair $(|\lambda|, \mu)$ is equal to
→
$\sin^{-1}(1-\text{x})-2\sin^{-1}\text{x}=\frac{\pi}{2}$
→

If a curve passes through the point $(1, -2)$ and has slope of the tangent at any point $(x,y)$ on it as $\frac{{{x^2} - 2y}}{x}$ then the curve also passes through the point

→
Evaluate: $\int_0^1 \frac{x \tan ^{-1} x}{\left(1+x^2\right)^{3 / 2}} d x$
→
Which of the following triplets give the direction cosines of a line:
→
The value of $\big[\vec{\text{a}}-\vec{\text{b}},\vec{\text{b}}-\vec{\text{c}},\vec{\text{c}}-\vec{\text{a}}\big],$ where $\big|\vec{\text{a}}\big|=1,\big|\vec{\text{b}}\big|=5,\big|\vec{\text{c}}\big|=3,$ is:
→
Let $R$ be the relation “is congruent to” on the set of all triangles in a plane is:
→
In a square matrix $A$ of order $3, a_{i i}'s$ are the sum of the roots of the equation $x^2 - (a + b)x + ab= 0$; $a_{i , i + 1}'s$ are the product of the roots, $a_{i , i - 1}'s$ are all unity and the rest of the elements are all zero. The value of the det. $(A)$ is equal to
→