Evaluate the following integrals: _ 6 ^ 3 ( + )^2dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}(\tan\text{x}+\cot\text{x})^2\text{dx}$

✓

Answer

$\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}(\tan\text{x}+\cot\text{x})^2\text{dx}$
$=\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}\big(\tan^2\text{x}+\cot^2\text{x}+2\tan\text{x }\cot\text{x})\text{dx}$
$=\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}\big(\sec^2\text{x}-1+\text{cosec}^2\text{x}-1+2\big)\text{dx}$
$=\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}\sec^2\text{x dx}+\int_{\frac{\pi}{6}}^\limits{\frac{\pi}{3}}\text{cosec}^2\text{x dx}$
$=\big[\tan\text{x}+(-\cot\text{x})\big]^{\frac{\pi}{3}}_\frac{\pi}{6}$
$=\Big(\tan\frac{\pi}{3}-\tan\frac{\pi}{6}\Big)-\Big(\cot\frac{\pi}{3}-\cot\frac{\pi}{6}\Big)$
$=\Big(\sqrt{3}-\frac{1}{\sqrt{3}}\Big)-\Big(\frac{1}{\sqrt{3}}-\sqrt{3}\Big)$
$=2\sqrt{3}-\frac{2}{3}$
$=\frac{4}{\sqrt{3}}$

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 INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the co-ordinates of the point where the line $\overrightarrow{r} = (\hat{-i} - 2\hat{j} -3\hat{k} + \lambda (3\hat{i} + 4\hat{j} + 3\hat{k})$ meets the plane which is perpendicular to the vector $\overrightarrow{n} = -\hat{i} - \hat{j} +3\hat{k} $ and at a distance of $\frac{4}{\sqrt{11}}$ from origin.

Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
f(x) = $\begin{matrix} \text{3x - 2}, && 02 \end{matrix}$.

Find the equation of the plane through the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and passing throught the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})=0$ and $\vec{\text{r}}\cdot(\hat{\text{j}}+2\hat{\text{k}})=0.$

Find $\frac{\text{dy}}{\text{dx}}$
$y = x^n + n^x + x^x + n^n$

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads $75\%$ of the times and third is also a biased coin that comes up tails $40\%$ of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Find the largest possible area of a right angled triangle whose hypotenuse is 5cm long.

Find $\text{y}=\text{Ae}^{-\text{kt}}\cos\text({pt}+\text{c})$prove that $\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}}+\text{n}^2\text{y}=0,$Where $\text{n}^2=\text{p}^2+\text{k}^2.$

If the product of the distances of the point (1, 1, 1) from the origin and the plane $x - y + z + λ = 0$ be 5, find the value of λ.

$\text{If y}=\text{e}^{\text{a}\cos^{-1}\text{x}},-1\leq\text{x}\leq1,\ \text{show that}(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{a}^2\text{y}=0.$

Find the area bounded by the parabola $x = 8 + 2y - y^2$, the y-axis and the line $y = -1$ and $y = 3$