Evaluvate the following intregals 2x+3 x^2+4x+5 dx - Vidyadip

Question

Evaluvate the following intregals
$\int\frac{2\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{2\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}$
Let $2\text{x}+3=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+4\text{x}+5)+\mu$
$=\lambda(2\text{x}+4)+\mu$
$2\text{x}+3=(2\lambda)\text{x}+4\lambda+\mu$
Compairing the coefficient of like powers of x,
$2\lambda=2\ \Rightarrow\lambda=1$
$4\lambda+\mu=3\Rightarrow4(1)+\mu=3$
$\Rightarrow\mu=-1$
So, $\text{I}=\int\frac{(2\text{x}+4)-1}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}$
$=\int\frac{(2\text{x}+4)}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}-\int\frac{1}{\sqrt{\text{x}^2+2\text{x}(2)+(2)^2-(2)^2+5}}$
$=\int\frac{(2\text{x}+4)}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}-\int\frac{1}{\sqrt{(\text{x}+2)^2+(1)^2}}\text{dx}$
$\text{I}=2\sqrt{\text{x}^2+4\text{x}+5}-\log\big|\text{x}+2+\sqrt{(\text{x}+2)^2+1}\big|+\text{C}$
$\Big[\text{since}, \int\frac{1}{\sqrt{\text{x}}}\text{dx}=2\sqrt{\text{x}}+\text{C},\int\frac{1}{\sqrt{\text{x}^2-\text{a}^2}}\text{dx}=\log\big|\text{x}+\sqrt{\text{x}^2-\text{a}^2}\big|+\text{C}\Big]$
$\text{I}=2\sqrt{\text{x}^2+4\text{x}+5}-3\log\big|\text{x}+2+\sqrt{\text{x}^2+4\text{x}+5}\big|+\text{C}$

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

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.

Determine graphically the minimum value of the objective function Z = -50x + 20y subject to the constraints:
2x - y $\geq$ - 5
3x + y $\geq$ 3
2x - 3y $\leq$ 12
x $\geq$ 0, y $\geq$ 0

Evaluate the following intregals:
$\int\frac{1}{5-4\cos\text{x}}\ \text{dx}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}=\text{at}^2,\text{y}=2\text{at at}\text{ t}=1$

How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit.

Solve the following differential equation
$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},\text{x}\neq0$

Prove that:
$\begin{vmatrix}\text{a}+\text{b}&\text{b}+\text{c}&\text{c}+\text{a}\\\text{b}+\text{c}&\text{c}+\text{a}&\text{a}+\text{b}\\\text{c}+\text{a}&\text{a}+\text{b}&\text{b}+\text{c}\end{vmatrix}=2\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{b}&\text{c}&\text{a}\\\text{c}&\text{a}&\text{b} \end{vmatrix}$

Let $A=\{1,2,3, \ldots .9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R(c, d)$ if $a+d=b+c$ for $(a, b),(c, d)$ in $A \times A$. Prove that R is an equivalence relation and also obtain the equivalence class $[(2,5)]$.

Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?

Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\sin\text{x dx}$