Integrate the function: 1 1+ x - Vidyadip

Question

Integrate the function: $\frac{1}{1+\cot x}$

✓

Answer

Let $I=\int \frac{1}{1+\cot x} d x$
$=\int \frac{1}{1+\frac{\cos x}{\sin x}} d x$
$=\int \frac{\sin x}{\sin x+\cos x} d x$
$= \frac{1}{2} \int \frac{(\sin x+\cos x)+(\sin x-\cos x)}{\sin x+\cos x} d x$
$=\frac{1}{2} \int 1 . d x+\frac{1}{2} \int \frac{(\sin x-\cos x)}{\sin x+\cos x} d x$
$= \frac{1}{2} x+\frac{1}{2} \int \frac{(\sin x-\cos x)}{\sin x+\cos x} d x$
Let sinx + cosx = t
$\Rightarrow \cos x-\sin x=\frac{d t}{d x}$
$\Rightarrow$ (cosx-sinx)dx = dt
$\Rightarrow -(\sin x-\cos x) dx = -dt$
Therefore, $I=\frac{x}{2}+\frac{1}{2} \int \frac{-d t}{t}$
$= \frac{x}{2}-\frac{1}{2} \log |t|+C$
$\Rightarrow I=\frac{x}{2}-\frac{1}{2} \log |\sin x+\cos x|+C$

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 "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

A diet of two foods $F_1$ and $F_2$ contains nutrients thiamine, phosphorous and iron.
The amount of each nutrient in each of the food $($in milligrams per $25$gms$) $is given in the following table:

Nutrients	Food	$F_1$	$F_2$
Thiamine		$0.25$	$0.10$
Phosphorous		$0.75$	$1.50$
Iron		$1.60$	$0.80$

The minimum requirement of the nutrients in the diet are $1.00$mg of thiamine, $7.50$mg of phosphorous and $10.00$mg of iron.
The cost of $F_1$ is $20$ paise per $25$gms while the cost of $F_2$ is 15 paise per 25gms.
Find the minimum cost of diet.

If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$, solve the system of linear equations:
$x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7$

Evaluate the following integrals:
$\int^\limits\frac{\pi}{2}_0\text{x}^2\sin\text{x dx}$

If $\text{A}=\begin{bmatrix}1&0\\-1&7\end{bmatrix},$ find k such that $A^2 - 8A + kI = 0.$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{ax}\text{ at }\Big(\frac{\text{a}}{\text{m}^2},\frac{2\text{a}}{\text{m}}\Big)$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = (\text{x}+\text{y})^2$

Find the angle between the follwing pairs of lines:$\vec{\text{r}}=\lambda\big(\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{j}}+\mu\big\{\big(\sqrt{3}-1\big)\hat{\text{i}}-\big(\sqrt{3}+1\big)\hat{\text{j}}+4\hat{\text{k}}\big\}$

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs. 300 and on one item of B is Rs. 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that,

Both balls are red,
First ball is black and second is red,
One of them is black and other is red.

Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$