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

Question

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

✓

Answer

Let $\text{I}=\int(\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$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, $2\lambda=1\Rightarrow\lambda=\frac{1}{2}$ $\lambda+\mu=1\Rightarrow\mu=\frac{1}{2}$ So, $\text{I}=\int\Big(\frac{1}{2}(2\text{x}+1)+\frac{1}{2}\Big)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$ $=\frac{1}{2}\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{dx}+\frac{1}{2}\int\sqrt{\text{x}^2+\text{x}+1}\text{dx}$ Let $\text{x}^2+\text{x}+1=\text{t}$ $\Rightarrow(2\text{x}+1)\text{dx = dt}$ $=\frac{1}{2}\int\sqrt{\text{t}}\text{dt}+\frac{1}{2}\int\sqrt{\Big(\text{x}+\frac{1}{2}\Big)^2+\Big(\frac{\sqrt3}{2}\Big)^2}\text{dx}$ $=\frac{1}{2}.\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\frac{1}{2}\begin{Bmatrix}\frac{\big(\text{x}+\frac{1}{2}\big)}{2}\sqrt{\text{x}^2+\text{x}+1}\\+\frac{3}{8}\log\bigg|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\bigg|\end{Bmatrix}+\text{C}$ $\Rightarrow\text{I}=\frac{1}{3}(\text{x}^2+\text{x}+1)^{\frac{3}{2}}+\frac{1}{8}(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\\+\frac{3}{16}\log\bigg|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\bigg|+\text{C}$ Hence, $\text{I}=\frac{1}{3}(\text{x}^2+\text{x}+1)^{\frac{3}{2}}+\frac{1}{8}(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\\+\frac{3}{16}\log\bigg|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\bigg|+\text{C}$

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

Using differentials, find the approximate values of the following:
$\sqrt{26}$

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Find the coordinates of a point on the parabola $y = x^{2 }+ 7x + 2$ which is closest to the strainght line $y = 3x -3.$

If $l_1, m_1, n_1; l_2, m_2, n_2; l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to $l_1 + l_2 + l_3, m_1 + m_2 + m_3, n_1 + n_2 + n_3$ makes equal angles with them.

Find the equation of the tangent to the curve $\text{x}=\sin3\text{t},\text{y}=\cos2\text{t}\text{ at }\text{t}=\frac{\pi}{4}$

By using properties of determinants, show that:
$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\2ab&1-a^2+b^2&2a\\2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$

Show that the binary operation $\ast \text{ on A = R - {-1}}$ defined as a $\text{a} \ast \text{b} = \text{a + b + ab}$ for all $\text{a, b}\in \text{A}$ is communicative and associative on A. Also find the identity element of $\ast$ in A and prove that every element of a is invertible.

Let $C$ be the set of all complex numbers and $C_{0 }$ be the set of all no$-$zero complex numbers. Let a relation $R$ on $C_{0 }$ be defined as $\text{z}_1\text{R z}_2\Leftrightarrow\frac{\text{z}_1-\text{z}_2}{\text{z}_1+\text{z}_2}$ is real for all $\text{z}_1,\ \text{z}_2\in\text{C}_0.$ Show that $R$ is an equivalence relation.

Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\cos\text{x}+\sin\text{x},\text{ y}\Big(\frac{\pi}{2}\Big)=1$

The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.