Integers are the set of numbers including all natural numbers and 0. They are part of the real numbers that do not include fractions, decimals, or negative numbers. Counting numbers are also considered whole numbers.

Natural numbers together with zero (0) are referred to as whole numbers. We know that natural numbers refer to the set of counting numbers starting from 1, 2, 3, 4 and so on. Simply put, integers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. Or we can say that integers are the set of non-negative integers.

Integers are the set of natural numbers togethe with the number 0. In mathematics, the set of integers is given as {0, 1, 2, 3, ...}, denoted by the symbol W.

W = {0, 1, 2, 3, 4, …}

All natural numbers are integers.

All integers are real numbers.

It is given that If n a whole number, and 0.01 is between 1/n and 1/n+2, then we need to find the value of n.

1/n+2 < 0.01 < 1/n

1/n+2 < 1/100 < 1/n

n< 100 < n +2

Put n = 99, then we get

99 < 100 < 101

Hence the value of n is 99.

Learn more about whole numbers here:

brainly.com/question/28052750

#SPJ4

3 0

2 years ago

Integrate e^x(sin(x) cos(x))

Karo-lina-s [1.5K]

$I=\int e^x(\sin(x)\cos(x))dx=\int e^x(\frac{1}{2}\sin(2x))dx=\frac{1}{2}\int e^x\sin(2x)dx$

$\text{If }u=\sin(2x)\to du=2\cos(2x)dx~\text{and}~dv=e^xdx\to v=e^x:\\\\
\text{Using }\int u\,dv=uv-\int v\,du:\\\\
I=\frac{1}{2}(e^x\sin(2x)-\int e^x(2\cos(2x))dx)\\\\
2I=e^x\sin(2x)-2\underbrace{\int e^x\cos(2x)dx}_{I_2}$

Looking for $I_2$ :

$\text{If}~u=\cos(2x)\to du=-2\sin(2x)dx~\text{and}~dv=e^xdx\to v=e^x:\\\\
I_2=e^x\cos(2x)-\int e^x(-2\sin(2x))dx\\\\ I_2=e^x\cos(2x)+2\int e^x(\sin(2x))dx\\\\ I_2=e^x\cos(2x)+2\int e^x(2\sin(x)\cos(x))dx\\\\ I_2=e^x\cos(2x)+4\int e^x(\sin(x)\cos(x))dx=e^x\cos(2x)+4I$

Replacing:

$2I=e^x\sin(2x)-2I_2\iff\\\\2I=e^x\sin(2x)-2(e^x\cos(2x)+4I)\iff\\\\
2I=e^x\sin(2x)-2e^x\cos(2x)-8I\iff\\\\
10I=e^x\sin(2x)-2e^x\cos(2x)\iff\\\\
I=\dfrac{e^x}{10}(\sin(2x)-2\cos(2x))\\\\
\boxed{\int e^x(\sin(x)\cos(x))dx=\dfrac{e^x}{10}(\sin(2x)-2\cos(2x))+C}$

6 0

4 years ago