1 / | | / 3 \ | | sin (x)| | |sin(x) - -------| dx | \ 3 / | / 0
Integral(sin(x) - sin(x)^3/3, (x, 0, 1))
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
There are multiple ways to do this integral.
Let .
Then let and substitute :
Integrate term-by-term:
The integral of is when :
The integral of a constant is the constant times the variable of integration:
The result is:
Now substitute back in:
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
Now substitute back in:
So, the result is:
The integral of sine is negative cosine:
The result is:
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
Now substitute back in:
So, the result is:
The integral of sine is negative cosine:
The result is:
So, the result is:
The integral of sine is negative cosine:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | | / 3 \ 3 | | sin (x)| 2*cos(x) cos (x) | |sin(x) - -------| dx = C - -------- - ------- | \ 3 / 3 9 | /
3 7 2*cos(1) cos (1) - - -------- - ------- 9 3 9
=
3 7 2*cos(1) cos (1) - - -------- - ------- 9 3 9
7/9 - 2*cos(1)/3 - cos(1)^3/9
Use the examples entering the upper and lower limits of integration.