Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{(\sin\text{x}-2\cos\text{x})(2\sin\text{x}-\cos\text{x})}\ \text{dx}$
✓

Answer

Let $\text{I}=\int\frac{1}{(\sin\text{x}-2\cos\text{x})(2\sin\text{x}-\cos\text{x})}\ \text{dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\Rightarrow\text{I}=\int\frac{\sec^2\text{x}}{\Big(\frac{\sin\text{x}-2\cos\text{x}}{\cos\text{x}}\Big)\times\Big(\frac{2\sin\text{x}+\cos\text{x}}{\cos\text{x}}\Big)}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{(\tan^2\text{x}-2)(2\tan\text{x}+1)}\ \text{dx}$
Let $\tan\text{x}=\text{t}$
$\Rightarrow\sec^2\text{x dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{(\text{t}-2)(2\text{t}-1 0)}$
$=\int\frac{\text{dt}}{2\text{t}^2+\text{t}-4\text{t}-2}$
$=\int\frac{\text{dt}}{2\text{t}^2-3\text{t}-2}$
$=\frac{1}{2}\int\frac{\text{dt}}{\text{t}^2-\frac{3}{2}\text{t}-1}$
$=\frac{1}{2}\int\frac{\text{dt}}{\text{t}^2-\frac{3}{2}\text{t}+\Big(\frac{3}{4}\Big)^2-\Big(\frac{3}{4}\Big)^2-1}$
$=\frac{1}{2}\int\frac{\text{dt}}{\Big(\text{t}-\frac{3}{4}\Big)-\frac{9}{16}-1}$
$=\frac{1}{2}\int\frac{\text{dt}}{\Big(\text{t}-\frac{3}{4}\Big)-\Big(\frac{5}{4}\Big)^2}$
$=\frac{1}{2}\times\frac{1}{2\times\frac{5}{4}}\log\Bigg|\frac{\text{t}-\frac{3}{4}-\frac{5}{4}}{\text{t}-\frac{3}{4}+\frac{5}{4}}\Bigg|+\text{C}$
$=\frac{1}{5}\ln\Bigg|\frac{\text{t}-2}{\text{t}+\frac{1}{2}}\Bigg|+\text{C}$
$=\frac{1}{5}\ln\Big|\frac{\text{t}-2}{2\text{t}+1}\Big|+\frac{1}{5}\int(2)+\text{C}$
$=\frac{1}{5}\ln\Big|\frac{\text{t}-2}{2\text{t}+1}\Big|+\text{C}$ where $\text{C}=\text{C}+\frac{1}{5}\int(2)$
$=\frac{1}{5}\ln\Big|\frac{\tan\text{x}-2}{2\tan\text{x}+1}\Big|+\text{C}$

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

$\text{if y } = \sqrt{\frac{(x - 3)(x^{2} + 4)}{3x^{2} + 4x + 5}}, \text{find} \frac{dy}{dx}$
→
Evaluate the following integrals:
$\int_{0}^\limits{1}\tan^{-1}\text{x dx}$
→
Find the value of $\lambda$ such that the line $\frac{\text{x}-2}{6}=\frac{\text{y}-1}{\lambda}=\frac{\text{z}+5}{-4}$ is perpendicular to the plane 3x - y - 2z = 7.
→
Find the vector equation of the line passing through the point (1, -1, 2) and perpendicular to the plane 2x - y + 3z - 5 = 0.
→
Find the distance between the lines l1 and l2 given by

$\vec{\text{r}}=\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}+\lambda\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$ and, $\vec{\text{r}}=3\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}+\mu\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$

→
Find the area of the region bounded by the paraola y2 = 4ax and line x = a.
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\frac{1}{1+\text{e}^{\tan\text{x}}}\text{ dx}$
→
Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 2,200. School Q wants to spend ₹ 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹ 1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.
→
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{e}^{-2\text{x}}\sin\text{x},\text{ y}(0)=0$
→
$\text{Evaluate} \int\limits^{3}_{1} (e^{2} - 3x + x^{2} + 1)\text{dx as a limit of a sum.} $
→