Evaluate the following integrals: (x+2) x^2+x+1 dx - Vidyadip

Question

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

✓

Answer

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

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the following for x and y.$\begin{bmatrix}3&-4\\9&2\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}10\\2\end{bmatrix}$

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.

Find the direction cosines of the line $\frac{4-\text{x}}{2}=\frac{\text{y}}{6}=\frac{1-\text{z}}{3}.$ Also, reduce it to vector form

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{\text{x}-3}{1}=\frac{\text{y}-6}{5}=\frac{\text{z}-4}{4}$

Show that $\text{f}\text{(x)}=\begin{cases}\frac{\sin 3\text{x}}{\tan2\text{x}},&\text{if } \text{x}<0\\\frac{3}{2},&\text{if }\text{x} = 0\\\frac{\log(1+3\text{x})}{\text{e}^{2\text{x}}},&\text{if}\text{ x}>0\end{cases}$ is discontinuous at x = 0.

Solve the following initial value problems $\tan\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)=2\text{x}\tan\text{x}+\text{x}^2-\text{y},\tan\text{x}\neq0$ given that y = 0 when $\text{x}=\frac{\pi}{2}$

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is:

Reflexive.
Symmetric.
Transitive.
All the three options.

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is$(a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{3}{2}}$

In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?

Show that $\text{AB}\neq\text{BA}$ in the following cases:
$\text{A}=\begin{bmatrix}1&3&-1\\2&-1&-1\\3&0&-1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}-2&3&-1\\-1&2&-1\\-6&9&-4\end{bmatrix}$