Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
$\int(\text{x}+2)\sqrt{3\text{x}+5}\text{ dx}$
✓

Answer

$\text{Let I}=\int(\text{x}+2)\sqrt{3\text{x}+5}\text{ dx}$
$\text{Putting}\ 3\text{x}+5=\text{t}$
$\Rightarrow\text{x}=\frac{\text{t}-5}{3}$
$\Rightarrow3\text{dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{3}$
$\therefore\text{I}=\int\Big(\frac{\text{t}-5}{3}+2\Big)\sqrt{\text{t}}\frac{\text{dt}}{3}$
$=\frac{1}{3}\int\Big(\frac{\text{t}-5+6}{3}\Big)\sqrt{t}\text{ dt}$
$=\frac{1}{9}\int(\text{t}+1)\sqrt{\text{t}}\text{ dt}$
$=\frac{1}{9}\int\Big(\text{t}^\frac{3}{2}+\text{t}^\frac{1}{2}\Big)\text{dt}$
$=\frac{1}{9}\bigg[\frac{2}{5}(3\text{x}+5)^\frac{5}{2}+\frac{2}{3}(3\text{x}+5)^\frac{3}{2}\bigg]+\text{C}$ $[\because\text{t}=3\text{x}+5]$
$=\frac{2}{9}\bigg[(3\text{x}+5)^\frac{3}{2}\Big\{\frac{3\text{x}+5}{5}+\frac{1}{3}\Big\}\bigg]+\text{C}$
$=\frac{2}{9}\bigg[(3\text{x}+5)^\frac{3}{2}\Big\{\frac{9\text{x}+15+5}{15}\Big\}\bigg]+\text{C}$
$=\frac{2}{9}\bigg[(3\text{x}+5)^\frac{3}{2}\Big\{\frac{9\text{x}+20}{15}\Big\}\bigg]+\text{C}$
$=\frac{2}{135}(3\text{x}+5)^\frac{3}{2}(9\text{x}+20)+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals:$\int\frac{\text{x}-3}{\text{x}^2+2\text{x}-4}\text{ dx}$
→
Find the angle between the following pairs of lines:
$\frac{\text{x}+4}{3}=\frac{\text{y}-1}{5}=\frac{\text{z}+3}{4}$ and $\frac{\text{x}+1}{1}=\frac{\text{y}-4}{1}=\frac{\text{z}-5}{2}$
→
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis twice abscissa of the pont of contact.
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = {x(x - 2)}^2$​​​​​​​
→
A man 160cm tall, walks away from a source of light situated at the top of a pole 6m high, at the rate of 1.1m/ sec. How fast is the length of his shadow increasing when he is 1m away from the pole?
→
If $\text{A}=\begin{bmatrix}4 & 3 \\ 2 & 5 \end{bmatrix},$ find x and y such that $A^2 = zA + yI = 0$. Hence, evaluate $A^{-1}.$
→
Differentiate the following functions with respect to x:
$\text{x}^{\text{x}\cos\text{x}}+\frac{\text{x}^2+1}{\text{x}^2-1}$
→
If the functions f(x), defined below is continuous at x = 0, find the value of k.
$\text{f(x)}=\begin{cases}\frac{1-\cos2\text{x}}{2\text{x}^2},&\text{x}<0\\\text{k},&\text{x}=0\\\frac{\text{x}}{|\text{x}|},&\text{x}>0\end{cases}$
→
Evaluate the following integrals:
$\int(\tan^{-1}\text{x}^2)\text{x dx}$
→
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?
→