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

Similar questions

Using mean value theorem, prove that there is a point on the curve $y = 2x^2 - 5x + 3$ between the points $A(1, 0)$ and $B(2, 1),$ where tangent is parallel to the chord AB. Also, find that point.

Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$ and
$\text{G}(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$
Show that
$\big[\text{F}(\alpha)\big]^{-1}=\text{F}(-\alpha)$

The pressure $p$ and the volume $v$ of a gas are connected by the relation $pv^{1.4} =$ const. Find the percentage error in $p$ corresponding to a decrease of $1/2\%$ in $v.$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}\text{ on }[0,\pi]$

If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:$\vec{\text{a}},\ \vec{\text{b}},\ 3\vec{\text{a}}-2\vec{\text{b}}$

The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&\text{if }0\leq\text{ x}<1\\\text{a},&\text{if }1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\text{if }\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous on $(0,\infty),$ then find the most suitable value of a and b.

Evaluate the following integrals:
$\int_{0}^\limits{\pi}\frac{1}{3+2\sin\text{x}+\cos\text{x}}\text{ dx}$

Find the equation of the plane through the points $(2, 2, -1)$ and $(3, 4, 2)$ and parallel to the line whose direction ratios are $7, 0, 6.$

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

If $\text{y}=\text{e}^{\tan^{-1}\text{x}},$ prove that $(1+\text{x}^2)\text{y}_2+(2\text{x}-1)\text{y}_1=0$