Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:
$\int(\text{x}=1)\sqrt{\text{x}^2-\text{x}+1}\text{dx}$
✓

Answer

Let $\text{I}=\int(\text{x}=1)\sqrt{\text{x}^2-\text{x}+1}\text{dx}\ \dots(1)$
Let $\text{x}+1=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2-\text{x}+1)+\mu$
$=\lambda(2\text{x}-1)+\mu$
Equating similar terms, we get,
$2\lambda=1\ \Rightarrow\ \lambda=\frac{1}{2}$
$-\lambda+\mu=1$
$\Rightarrow\mu=1+\lambda=1+\frac{1}{2}=\frac{3}{2}$
$\therefore\ \mu=\frac{3}{2}$
So,
$\text{I}=\int\Big(\frac{1}{2}(2\text{x}-1)+\frac{3}{2}\Big)\sqrt{\text{x}^2-\text{x}+1}\text{dx}$
$=\frac{1}{2}\int(2\text{x}-1)\sqrt{\text{x}^2-\text{x}+1}\text{dx}+\frac{3}{2}\int\sqrt{\text{x}^2-\text{x}+1}\text{dx}$
Let $\text{x}^2-\text{x}+1=\text{t}$
$\Rightarrow(2\text{x}-1)\text{dx}=\text{dt}$
$=\frac{1}{2}\int\sqrt{\text{t}}\text{dt}+\frac{3}{2}\int\sqrt{\Big(\text{x}-\frac{1}{2}\Big)^2+\Big(\frac{\sqrt3}{2}\Big)^2}\text{dx}$
$=\frac{1}{2}\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\frac{3}{2}\begin{Bmatrix}\frac{\big(\text{x}-\frac{1}{2}\big)}{2}\sqrt{\text{x}^2-\text{x} +1}\\+\frac{3}{8}\log\Big|\Big(\text{x}-\frac{1}{2}\Big)+\sqrt{\text{x}^2-\text{x}+1}\Big|\end{Bmatrix}$
$\Rightarrow\text{I}=\frac{1}{3}\text{t}^{\frac{3}{2}}+\frac{3}{8}(2\text{x}-1)\sqrt{\text{x}^2-\text{x}+1}+\frac{9}{16}\\\times\log\Big|\Big(\text{x}-\frac{1}{2}\Big)+\sqrt{\text{x}^2-\text{x}+1}\Big|+\text{C}$
Hence,
$\Rightarrow\text{I}=\frac{1}{3}(\text{x}^2-\text{x}+1)^{\frac{3}{2}}+\frac{3}{8}(2\text{x}-1)\sqrt{\text{x}^2-\text{x}+1}+\frac{9}{16}\\\times\log\Big|\Big(\text{x}-\frac{1}{2}\Big)+\sqrt{\text{x}^2-\text{x}+1}\Big|+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.
→
Evaluate the following:
$\begin{bmatrix}1&2&3\end{bmatrix}\begin{bmatrix}1&0&2\\2&0&1\\0&1&2\end{bmatrix}\begin{bmatrix}2\\4\\6\end{bmatrix}$
→
Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the mean and variance of the number of red cards.
→
Solve the following differential equation:
$(\text{x}+\tan\text{y})\text{dy}=\sin2\text{y dx}$
→
Evaluate the following integrals:

$\int\frac{1}{\cos\text{x}+\text{cosec x}}\text{dx}$

→
Solve the following differential equation:

$\sqrt{\text{1 + x}^{2}+\text{y}^{2}+\text{x}^{2}\text{y}^{2}}+\text{xy}\frac{\text{dy}}{\text{dx}}=0.$

→
Verify Rolle's theorem for the following function on the indicated intervals

$\text{f}(\text{x})=\sin2\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$

→
If AB = BA for any two sqaure matrices, prove by mathematical induction that (AB)n = AnBn.
→
Find the image of the point (0, 0, 0) in the plane 3x + 4y - 6z + 1 = 0.
→
If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is:
  1. {1, 4, 6, 9}
  2. {4, 6, 9}
  3. {1}
  4. None of these.
→