Evaluvate the following intregals 2x+1 x^2+4x+3 dx - Vidyadip

Question

Evaluvate the following intregals
$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$
Let $2\text{x}+1=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+4\text{x}+3)+\mu$
$=\lambda(2\text{x}+4)+\mu$
$2\text{x}+1=(2\lambda)\text{x}+4\lambda+\mu$
Compairing the coefficient of like powers of x,
$2\lambda=2\ \Rightarrow\lambda=1$
$4\lambda+\mu=1\Rightarrow4(1)+\mu=1$
$\Rightarrow\mu-3$
So, $\text{I}=\int\frac{(2\text{x}+4)-3}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$
$=\int\frac{(2\text{x}+4)}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}-3\int\frac{1}{\sqrt{\text{x}^2+2\text{x}(2)+(2)^2-(2)^2+3}}$
$=\int\frac{(2\text{x}+4)}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}-3\int\frac{1}{\sqrt{(\text{x}+2)^2-1}}\text{dx}$
$\text{I}=2\sqrt{\text{x}^2+4\text{x}+3}-3\log\big|\text{x}+2+\sqrt{(\text{x}+2)^2-1}\big|+\text{C}$
$\big[\text{since}, \int\frac{1}{\sqrt{\text{x}}}\text{dx}=2\sqrt{\text{x}}+\text{C},\int\frac{1}{\sqrt{\text{x}^2-\text{a}^2}}\text{dx}=\log\big|\text{x}+\sqrt{\text{x}^2-\text{a}^2}\big|+\text{C}$
$\text{I}=2\sqrt{\text{x}^2+4\text{x}+3}-3\log\big|\text{x}+2+\sqrt{\text{x}^2+4\text{x}+3}\big|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove the following identities:
$\begin{vmatrix}\text{x}+\lambda&2\text{x}&2\text{x}\\2\text{x}&\text{x}+\lambda&2\text{x}\\2\text{x}&2\text{x}&\text{x}+\lambda\end{vmatrix}$
$=(5\text{x}+\lambda)(\lambda-\text{x})^2$

A manufacturer produces two items A and B. Both are processed on two machines I and II. A needs 2 hours on machine I and 2 hours on machine II. B needs 3 hours on machine I and 1 hour on machine II. If machine I can run maximum 12 hours per day and II for 8 hours per day, construct a problem in the form of inequations and find its feasible solution graphically.

Find the distance of the point with position vector $-\hat{\text{i}}-5\hat{\text{j}}-10\hat{\text{k}}$ from the point of intersection of the line $\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+12\hat{\text{k}})$ with the plane $\vec{\text{r}}.(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5.$

Evaluate the following integrals:
$\int\frac{\cos^5\text{x}}{\sin\text{x}}\text{ dx}$

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from,
LONDON.

Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

If $\text{A}=\begin{bmatrix}2 & 3 \\ 1 & 2 \end{bmatrix},$ verify that $A^2 - 4A + I = 0,$ where $\text{I}=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\text{ and O}\begin{bmatrix}0 & 0 \\ 0 & 0 \end{bmatrix}.$ Hence find $A^{-1}.$

Evaluate : $\int \frac{\cos \theta}{\cos 3 \theta} \cdot d \theta$

Compute the adjoint of the following matrices:$\begin{bmatrix}2 & -1 & 3 \\4 & 2 & 5 \\ 0 & 4 & -1 \end{bmatrix}$
Verify that (adj A)A = |A|I = A (adj A) for the above matrices.