Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 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 "5 Marks Questions" 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

Find the equation of the plane which contains two parallel lines $\frac{\text{x}-4}{1}=\frac{\text{y}-3}{-4}=\frac{\text{z}-2}{5}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}+2}{-4}=\frac{\text{z}}{5}.$
→
Prove that the points $\hat{\text{i}}-\hat{\text{j}},\ 4\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$ and $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ are the vertices of a right-angled triangle.
→
R is a relation on the set Z of integers and it is given by (x, y) ∈ R ⇔ | x - y | ≤ 1. Then, R is:
  1. Reflexive and transitive.
  2. Reflexive and symmetric.
  3. Symmetric and transitive.
  4. An equivalence relation.
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\log\text{x}$
→
The contents of urns I, II, III are as follows:
Urn I : 1 white, 2 black and 3 red balls
Urn II : 2 white, 1 black and 1 red balls
Urn III : 4 white, 5 black and 3 red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns I, II, III?
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$
→
If $\text{A}=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix},$ show that $A^2 - 7A + 10I_3 = 0$.
→
Solve the following system of equations by matrix method:
$8x + 4y + 3z = 18$
$2x + y + z = 5$
$x + 2y + z = 5$
→
Two dice are thrown together and the total score is noted. The event E, F and G are "a total of 4", "a total of 9 or more", and "a total divisible by 5", respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.
→
By Using properties of definite integrals, evaluate the following integral in Exercise:
$\int^{\pi}_{0}\log(1+\cos\text{x})\text{dx}$
→