Question
Evaluate the following integrals:$\int\frac{(\text{x}-1)^2}{\text{x}^2+2\text{x}+2}\text{ dx}$
✓
Answer
Let $\int\frac{(\text{x}-1)^2}{\text{x}^2+2\text{x}+2}\text{ dx}$ $=\int\Big(\frac{\text{x}^2-2\text{x}+1}{\text{x}^2-2\text{x}+2}\Big)\text{dx}$ Here,
Therefore,
$\frac{\text{x}^2-2\text{x}+1}{\text{x}^2+2\text{x}+2}=1-\frac{(4\text{x}+1)}{\text{x}^2+2\text{x}+2}\ ....(1)$ Let $4\text{x}+1=\text{A}\frac{\text{d}}{\text{dx}}\big(\text{x}^2+2\text{x}+2\big)+\text{B}$ $4\text{x}+1=\text{A}(2\text{x}+2)+\text{B}$ $4\text{x}+1=(2\text{A})\text{x}+2\text{A}+\text{B}$ Equating Cofficients of like terms $2\text{A}=4$ $\text{A}=2$ $2\text{A}+\text{B}=1$ $2\times2+\text{B}=1$ $\text{B}=-3$ $\int\Big(\frac{\text{x}^2-2\text{x}+1}{\text{x}^2-2\text{x}+2}\Big)\text{dx}$ $=\int\text{dx}-2\int\frac{(2\text{x}+2)}{\text{x}^2+2\text{x}+2}\text{ dx}+3\int\frac{\text{ dx}}{\text{x}^2+2\text{x}+2}$ $=\int\text{dx}-2\int\frac{(2\text{x}+2)}{\text{x}^2+2\text{x}+2}\text{ dx}+3\int\frac{\text{dx}}{(\text{x}+1)^2+1^2}$ $=\text{x}-2\log\big|\text{x}^2+2\text{x}+2\big|+\frac{3}{1}\tan^{-1}\Big(\frac{\text{x}+1}{1}\Big)+\text{C}$ $=\text{x}-2\log\big|\text{x}^2+2\text{x}+2\big|+3\tan^{-1}(\text{x}+1)+\text{C}$
Therefore,
$\frac{\text{x}^2-2\text{x}+1}{\text{x}^2+2\text{x}+2}=1-\frac{(4\text{x}+1)}{\text{x}^2+2\text{x}+2}\ ....(1)$ Let $4\text{x}+1=\text{A}\frac{\text{d}}{\text{dx}}\big(\text{x}^2+2\text{x}+2\big)+\text{B}$ $4\text{x}+1=\text{A}(2\text{x}+2)+\text{B}$ $4\text{x}+1=(2\text{A})\text{x}+2\text{A}+\text{B}$ Equating Cofficients of like terms $2\text{A}=4$ $\text{A}=2$ $2\text{A}+\text{B}=1$ $2\times2+\text{B}=1$ $\text{B}=-3$ $\int\Big(\frac{\text{x}^2-2\text{x}+1}{\text{x}^2-2\text{x}+2}\Big)\text{dx}$ $=\int\text{dx}-2\int\frac{(2\text{x}+2)}{\text{x}^2+2\text{x}+2}\text{ dx}+3\int\frac{\text{ dx}}{\text{x}^2+2\text{x}+2}$ $=\int\text{dx}-2\int\frac{(2\text{x}+2)}{\text{x}^2+2\text{x}+2}\text{ dx}+3\int\frac{\text{dx}}{(\text{x}+1)^2+1^2}$ $=\text{x}-2\log\big|\text{x}^2+2\text{x}+2\big|+\frac{3}{1}\tan^{-1}\Big(\frac{\text{x}+1}{1}\Big)+\text{C}$ $=\text{x}-2\log\big|\text{x}^2+2\text{x}+2\big|+3\tan^{-1}(\text{x}+1)+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Verify Rolle's theorem for the following function on the indicated intervals $f(x) = (x - 1) (x - 2)^2$ on $[1, 2]$
→Let C be a curve defined parametrically as $\text{x}=\text{a}\cos^3\theta,\text{y}=\text{a}\sin^3\theta,0\leq\theta\leq\frac{\pi}{2}.$ Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).
→Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
Evaluate :
→$\int_0^1 \frac{\log x}{\sqrt{1-x^2}} \cdot d x$
Find the area enclosed by the parabola $y = 5x^2$ and $y = 2x^2 + 9.$
→Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
Solve the following system of equations by matrix method:
$x + y + z = 6$
$x + 2z = 7$
$3x + y + z = 12$
→$x + y + z = 6$
$x + 2z = 7$
$3x + y + z = 12$
Find the vector equation of the plane passing through the points (1, 1, -1), (6, 4, -5) and (-4, -2, 3).
Prove that the function
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}}{|\text{x|+2}\text{x}^2}, &\text{ x}\neq0\\\text{k}, &\text{ x}=0\end{cases}$
remains discontinuous at x = 0, regardless the choice of k.
→$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}}{|\text{x|+2}\text{x}^2}, &\text{ x}\neq0\\\text{k}, &\text{ x}=0\end{cases}$
remains discontinuous at x = 0, regardless the choice of k.
Differentiate $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big)$ with respect to $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$
→