Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:

$\int\frac{5\text{x}-2}{1+2\text{x}+3\text{x}^2}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{5\text{x}-2}{1+2\text{x}+3\text{x}^2}\text{ dx}$

Rewriting the numerator we have,

$5\text{x}-2=\text{A}\frac{\text{d}}{\text{dx}}\big(1+2\text{x}+3\text{x}^2\big)+\text{B}$

$\Rightarrow5\text{x}-2=\text{A}(2+6\text{x})+\text{B}$

$\Rightarrow5\text{x}-2=6\text{xa}+2\text{A}+\text{B}$

Comparing the coefficient, we have,

$6\text{A}=5$ and $2\text{A}+\text{B}=-2$

$\Rightarrow\text{A}=\frac{5}{6}$

Substituting the value of A in 2A + B = -2, we have,

$2\times\frac{5}{6}+\text{B}=-2$

$\Rightarrow\frac{10}{6}+\text{B}=-2$

$\Rightarrow\text{B}=-2-\frac{10}{6}$

$\Rightarrow\text{B}=\frac{-12-10}{6}$

$\Rightarrow\text{B}=\frac{-22}{6}$

$\Rightarrow\text{B}=\frac{-11}{3}$

$5\text{x}-2=\frac{5}{6}(2+6\text{x})-\frac{11}{3}$

Thus, $\text{I}=\int\frac{5\text{x}-2}{1+2\text{x}+3\text{x}^2}\text{ dx}$ becomes,

$\text{I}=\int\frac{\big[\frac{5}{6}(2+6\text{x})-\frac{11}{3}\big]}{3\text{x}^2+2\text{x}+1}\text{ dx}$

$=\frac{5}{6}\int\frac{(2+6\text{x})}{3\text{x}^2+2\text{x}+1}\text{ dx}-\frac{11}{3}\int\frac{\text{dx}}{3\text{x}^2+2\text{x}+1}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{3\times3}\int\frac{\text{dx}}{\text{x}^2+\frac{2}{3}\text{x}+\frac{1}{3}}+\text{C}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{9}\int\frac{\text{dx}}{\text{x}^2+\frac{2}{3}\text{x}+\big(\frac{4}{3}\big)^2+\frac{1}{3}-\big(\frac{4}{3}\big)^2}+\text{C}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{9}\int\frac{\text{dx}}{\big(\text{x}+\frac{1}{3}\big)^2+\Big(\frac{\sqrt2}{3}\Big)^2}+\text{C}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{9}\times\frac{1}{\frac{\sqrt2}{3}}\tan^{-1}\Bigg[\frac{\big(\text{x}+\frac{1}{3}\big)}{\frac{\sqrt2}{3}}\Bigg]+\text{C}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{9}\times\frac{3}{\sqrt2}\tan^{-1}\Bigg[\frac{\big(\frac{3\text{x}+1}{3}\big)}{\frac{\sqrt2}{3}}\Bigg]+\text{C}$

$=\frac{5}{6}\log\big(3\text{x}^2+2\text{x}+1\big)-\frac{11}{3\sqrt2}\tan^{-1}\Big[\frac{3\text{x}+1}{\sqrt2}\Big]+\text{C}$

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 "4 Marks" 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

If $\vec{\text{a}} \times \vec{\text{b}} = \vec{\text{c}} \times \vec{\text{d}}$ and $\vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \times \vec{\text{d}},$ show that $\vec{\text{a}} - \vec{\text{d}}$ is parallel to $\vec{\text{b}} - \vec{\text{c}},$ where $\vec{\text{a}} \neq \vec{\text{d}}$ and $\vec{\text{b}} \neq \vec{\text{c}}.$
→
A die is thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.
→
If ex + xy = ex+y, prove that $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$
→
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm - 4ln + mn = 0
→
Find the distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z.
→
The two adjacent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}-5\hat{\text{k}}$ and $2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}.$ Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.
→
Solve the following LPP graphically:
Minimise Z = 3x + 9y
subject to the constraints
x + 3y $\leq$ 60
x + y $\geq$ 10
x $\leq$ y
x $\geq$ 0, y $\geq$ 0.
→
If $\text{x}\sin(\text{a}+\text{y})+\sin\text{a}\cos(\text{a}+\text{y})=0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin\text{a}}$
→
Solve the following systems of homogeneous linear equations by matrix method:
2x - y + 2z = 0
5x + 3y - z = 0
x + 5y - 5z = 0
→
Evaluate the following integrals:

$\int\frac{2\text{x}-3}{\text{x}^2+6\text{x}+13}\text{ dx}$

→