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 "4 Marks" 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

Evaluate the following intregals:
$\int\frac{1}{5-4\sin\text{x}}\ \text{dx}$

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi:(\pi+2)$.

Evaluate:
$\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$

Prove that:
$\begin{vmatrix}\text{x}+4&\text{x}&\text{x}\\\text{x}&\text{x}+4&\text{x}\\\text{x}&\text{x}&\text{x}+4\end{vmatrix}=16(3\text{x}+4)$

Let n be a fixed positive integer. Define a relation R on Z as follows:
$(\text{a, b})\in\text{R}\Leftrightarrow\ \text{a}-\text{b}$ is divisible by n. Show that R is an equivalence relation on Z.

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 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 equation of the plane which bisects the line segment joining the points (-1, 2, 3) and (3, -5, 6) at right angles.

Show that the following curves intersect orthogonally at the indicated points:
x² = 4y and 4y + x² = 8 at (2, 1)

Show that four points whose position vectors are
$6\hat{\text{i}}-7\hat{\text{j}},16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},3\hat{\text{i}}-6\hat{\text{k}},2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

Differentiate the following functions with respect to x:
$\frac{2^\text{x}\cos\text{x}}{(\text{x}^2+3)^2}$