1 / | | / 2 3 \ | \sin(x) + sin (x) + sin (x)/ dx | / 0
Integral(sin(x) + sin(x)^2 + sin(x)^3, (x, 0, 1))
Integrate term-by-term:
Integrate term-by-term:
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant is the constant times the variable of integration:
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 cosine is sine:
So, the result is:
Now substitute back in:
So, the result is:
The result is:
The integral of sine is negative cosine:
The result is:
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:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | 3 | / 2 3 \ x sin(2*x) cos (x) | \sin(x) + sin (x) + sin (x)/ dx = C + - - 2*cos(x) - -------- + ------- | 2 4 3 /
3 13 cos (1) cos(1)*sin(1) -- - 2*cos(1) + ------- - ------------- 6 3 2
=
3 13 cos (1) cos(1)*sin(1) -- - 2*cos(1) + ------- - ------------- 6 3 2
13/6 - 2*cos(1) + cos(1)^3/3 - cos(1)*sin(1)/2
Use the examples entering the upper and lower limits of integration.