x^2+5x+2 x+2 dx - Vidyadip

Question

$\int\frac{\text{x}^2+5\text{x}+2}{\text{x}+2}\text{dx}$

✓

Answer

$\int\frac{(\text{x}^2+5\text{x}+2)}{(\text{x}+2)}\text{dx}$
$=\int\frac{\text{x}^2}{\text{x}+2}\text{dx}+5\int\frac{\text{x dx}}{\text{x}+2}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int\Big(\frac{\text{x}^2-4+4}{\text{x}+2}\Big)\text{dx}+5\int\Big(\frac{\text{x}+2-2}{\text{x}+2}\Big)\text{dx}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int\frac{(\text{x}-2)(\text{x}+2)}{(\text{x}+2)}\text{dx}+\int\frac{4}{\text{x}+2}\text{dx}+5\int\Big(1-\frac{2}{\text{x}+2}\Big)\text{dx}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int(\text{x}-2)\text{dx}+4\int\frac{\text{dx}}{\text{x}+2}+5\int\text{dx}-10\int\frac{\text{dx}}{\text{x}+2}+2\int\frac{\text{dx}}{\text{x}+2}$
$=\int(\text{x}-2)\text{dx}-4\int\frac{\text{dx}}{\text{x}+2}+5\int\text{dx}$
$=\Big(\frac{\text{x}^2}{2}-2\text{x}\Big)-4\text{ln|}\text{x}+2|+5\text{x}+\text{C}$
$=\frac{\text{x}^2}{2}+3\text{x}-4\text{ln}|\text{x}+2|+\text{C}$

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

Similar questions

Prove the following using properties of determinants:
$ \begin{vmatrix} \text{a + b + 2c} & \text{a} & \text{b} \\ \text{c} & \text{b + c + 2a} & \text{b} \\ \text{c} & \text{a} & \text{c + a + 2b} \end{vmatrix}= 2(\text{a + b + c})^3 $

If A + B + C = 0, then prove that $\begin{vmatrix}1&\cos\text{C}&\cos\text{B}\\\cos\text{C}&1&\cos\text{A}\\\cos\text{B}&\cos\text{A}&1\end{vmatrix}=0.$

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

Find matrix A such that
$\begin{bmatrix} 2& -1\$0.3em] 1 & 0 \$0.3em] -3 & 4 \end{bmatrix}\text{A}=\begin{bmatrix} -1& -8\$0.3em] 1 & -2 \$0.3em] 9 & 22 \end{bmatrix}$

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

Find the point on the curve $x^2= 8y$ which is nearest to the point $(2,4).$

Show that the cone of the greatest volume which can be inscribed in a given spher has an altitude equal to 2/3 of the diameter of the sphere.

Using the proprties of determinants in Exercise:
$\begin{vmatrix}\text{y}^2\text{z}^2&\text{yz}&\text{y}+\text{z}\\\text{z}^2\text{x}^2&\text{zx}&\text{z}+\text{x}\\\text{x}^2\text{y}^2&\text{xy}&\text{x}+\text{y}\end{vmatrix}=0$

If $\text{y}=\cos^{-1}\Big\{\frac{2\text{x}-3\sqrt{1-\text{x}^2}}{\sqrt{13}}\Big\},$ find $\frac{\text{dy}}{\text{dx}}.$

In a hockey match, both teams $A$ and $B$ scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.