Evaluate the following integrals: ^ 3 _ 6 1 1+ ^ 3 2 x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}\ ...(\text{i})$
Then,
$\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\big(\frac{\pi}{3}+\frac{\pi}{6}-\text{x}\big)}\text{ dx}$ $\Bigg[\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_\text{a}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\big(\frac{\pi}{2}-\text{x}\big)}\text{ dx}$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\tan^{\frac{3}{2}}\text{x}}\text{ dx}$
$=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{\cot^{\frac{3}{2}}\text{x}}{\cot^{\frac{3}{2}}\text{x}+1}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii)
$2\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1+\cot^{\frac{3}{2}}\text{x}}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\text{dx}$
$\Rightarrow2\text{I}=\big[\text{x}\big]^{\frac{\pi}{3}}_\frac{\pi}{6}$
$\Rightarrow2\text{I}=\frac{\pi}{3}-\frac{\pi}{6}=\frac{\pi}{6}$
$\Rightarrow\text{I}=\frac{\pi}{12}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(2\text{x}+1)(\text{x}^2-1)}\text{ dx}$

Determine whether the following pair of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$

An amount of $Rs. 10,000$ is put into three investments at the rate of $10, 12$ and $15\%$ per annum. The combined income is $Rs. 1310$ and the combined income of first and second investment is $Rs. 190$ short of the income from the third. Find the investment in each using matrix method.

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission.

Month	Sale of units			Total commission drawn (in Rs.)
	A	B	C
Jan	90	100	20	800
Feb	130	50	40	900
March	60	100	30	850

Find out the rates of commission on items A, B and C by using determinant method.

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

Find the intervals in which f(x) is increasing or decreasing:
$\text{f}(\text{x})=\sin\text{x}(1+\cos\text{x}),0<\text{x}<\frac{\pi}{2}$

$\text{if} \overrightarrow{\text{r}} = x\hat{\text{i}} + y\hat{\text{j}} + z\hat{\text{k}}, \text{find} \overrightarrow(\text{r} \times \hat{\text{i}}). (\overrightarrow{\text{r}} \times \text{j}) + xy$

Prove using vector: the quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

The total area of a page is $150\ cm^2.$ The combined width of the margin at the top and bottom is $3\ cm$ and the side $2\ cm.$ What must be the dimensions of the page in order that the area of the printed matter may be maximum?

In a hockey match, both teams $A$ and $B$ scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team $A$ was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.