Evaluate the following intregals: 5x^2+20x+6 x^2+2x^2+x dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}^2+2\text{x}^2+\text{x}}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}^2+2\text{x}^2+\text{x}}\ \text{dx}$$=\int\frac{5\text{x}^2+20\text{x}+6}{\text{x}(\text{x}+1)^2}\ \text{dx}$
Now,
Let $\frac{5\text{x}^2+20\text{x}+6}{\text{x}(\text{x}+1)^2}=\frac{\text{A}}{\text{x}}+\frac{\text{B}}{\text{x}+1}+\frac{\text{C}}{(\text{x}+1)^2}$
$\Rightarrow5\text{x}^2+20\text{x}+6=\text{A}(\text{x}+1)^2+\text{Bx}(\text{x}+1)+\text{Cx}$
Equating similar terms, we get,
A + B = 5, 2A + B + C = 20, A = 6
Solving, we get, B = -1, C = 9
Thus,
$\text{I}=\int\frac{6\text{dx}}{\text{x}}-1\int\frac{\text{dx}}{\text{x}+1}+9\int\frac{\text{dx}}{(\text{x}+1)^2}$
$\therefore\text{I}=6\log|\text{x}|-\log|\text{x}+1|-\frac{9}{\text{x}+1}+\text{C}$

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

If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$

If $x^x + y^x = 1,$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}(\text{y}+\text{x}\log\text{y})}{\text{x}(\text{y}\log\text{x}+\text{x})}$

Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{k}})+\lambda\hat{\text{i}}+\mu(\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}})$

$\text{Evaluate:} \int\limits_0^\frac{\pi}{2} (2\log \sin \text{x} - \log \sin 2\text{x}) \text{dx}$

Find the vector equation of the plane passing through the points (1, 1, 1), (1, -1, 1) and (-7, -3, -5)

Evaluate the following integrals:
$\int\cos^{-1}(4\text{x}^3-3\text{x})\text{dx}$

Find a unit vector perpendicular to each of the vectors $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}},$ where $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}.$

Evaluate:
$\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$

Let $R$ be relation defined on the set of natural number $N$ as follows: $R=\{(x, y): x \in N, y \in N, 2 x+y=41\}$. Find the domain and range of the relation $R$. Also verify whether $R$ is reflexive, symmetric and transitive.

If the lines $\text{x}=5,\frac{\text{y}}{3-\alpha}=\frac{\text{z}}{-2}$ and $\text{x}=\alpha,\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha}$ are coplanar, find the values of $\alpha.$