Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\frac{2\text{x}}{2+\text{x}-\text{x}^2}\text{ dx}$
✓

Answer

$\int\frac{2\text{x}\text{ dx}}{(2+\text{x}-\text{x}^2)}$ $2\text{x}=\text{A}\frac{\text{d}}{\text{dx}}\big(2+\text{x}-\text{x}^2\big)+\text{B}$ $2\text{x}=\text{A}(0+1-2\text{x})+\text{B}$ $2\text{x}=(-2\text{A})\text{x}+\text{A}+\text{B}$ Comparing the coefficients of like power of x, $-2\text{A}=2$ $\text{A}=-1$ $\text{A}+\text{B}=0$ $-1+\text{B}=0$ $\text{B}=1$ Now, $\int\frac{2\text{x}\text{ dx}}{(2+\text{x}-\text{x}^2)}$ $=\int\Big(\frac{-1(1-2\text{x})+1}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$ $=-\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}+\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$ $=-\text{I}_1+\text{I}_2\ ....(1)$ (say) where $\text{I}_1=\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$ $\text{I}_2=\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$ $\text{I}_1=\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$ Let $-\text{x}^2+\text{x}+2=\text{t}$ $\Rightarrow(1-2\text{x})\text{dx}=\text{dt}$ $\text{I}_1=\int\frac{\text{dt}}{\text{t}}$ $\text{I}_1=\log|\text{t}|+\text{C}_1\ ....(2)$ $\text{I}_2=\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$ $\text{I}_2=\int\frac{-\text{dx}}{\text{x}^2+\text{x}+2}$ $\text{I}_2=\int\frac{-\text{dx}}{\text{x}^2-\text{x}+\big(\frac{1}{2}\big)^2-\big(\frac{1}{2}\big)^2-2}$ $\text{I}_2=\int\frac{-\text{dx}}{\big(\text{x}-\frac{1}{2}\big)^2-\big(\frac{3}{2}\big)^2}$ $\text{I}_2=-\frac{1}{2\times\frac{3}{2}}\log\Bigg|\frac{\text{x}-\frac{1}{2}-\frac{3}{2}}{\text{x}-\frac{1}{2}+\frac{3}{2}}\Bigg|+\text{C}_2$ $\text{I}_2=-\frac{1}{3}\log\Big|\frac{\text{x}-2}{\text{x}+1}\Big|+\text{C}_2\ ....(3)$ From (1) (2) and (3) $\int\Big(\frac{2\text{x}}{2+\text{x}+\text{x}^2}\Big)\text{ dx}=-\log\big|2+\text{x}-\text{x}^2\big|-\frac{1}{3}\log\Big|\frac{\text{x}-2}{\text{x}+1}\Big|+\text{C}_1+\text{C}_2$ $=-\log\big|2+\text{x}-\text{x}^2\big|+\frac{1}{3}\log\Big|\frac{1+\text{x}}{\text{x}-2}\Big|+\text{C}$Where, $\text{C}=\text{C}_1+\text{C}_2$

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

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=2\cos\text{x},\text{ y}\Big(\frac{\pi}{2}\Big)=0$
→
Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}+\text{y}=\tan^{-1}\text{x}$
→
Without expanding, prove that:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\\\text{a}&\text{b}&\text{c}\end{vmatrix}=\begin{vmatrix}\text{y}&\text{b}&\text{q}\\\text{x}&\text{a}&\text{p}\\\text{z}&\text{c}&\text{r}\end{vmatrix}$
→
Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+3\hat{\text{i}}+4\hat{\text{k}}.$
→
Show that a closed right circular cylinder of a given surface area has maximum volume if its height equals the diameter of its base.
→
Evaluate the following integrals:$\int\limits^2_{-1}\big(|\text{x}+1|+|\text{x}|+|\text{x}-1|\big)\text{dx}$
→
Solve the following systems of linear equations by cramer's rule:
5x + 7y = -2
4x + 6y = -3
→
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
→
Evaluate the following integrals:
$\int\frac{\text{x}^2+1}{\text{x}^4-\text{x}^2+1}\ \text{dx}$
→
Differentiate the following functions with respect to x:
$(\log\text{x})^\text{x}$
→