1 / | | 2 3 | x *cos (x) dx | / 0
Integral(x^2*cos(x)^3, (x, 0, 1))
Use integration by parts:
Let and let .
Then .
To find :
Rewrite the integrand:
There are multiple ways to do this integral.
Let .
Then let and substitute :
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:
The integral of is when :
So, the result is:
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 is when :
Now substitute back in:
So, the result is:
The integral of cosine is sine:
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 is when :
Now substitute back in:
So, the result is:
The integral of cosine is sine:
The result is:
Now evaluate the sub-integral.
Use integration by parts:
Let and let .
Then .
To find :
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
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:
So, the result is:
The integral of a constant times a function is the constant times the integral of the function:
The integral of sine is negative cosine:
So, the result is:
The result is:
Now evaluate the sub-integral.
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Let .
Then let and substitute :
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:
The integral of is when :
So, the result is:
The result is:
Now substitute back in:
So, the result is:
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:
The result is:
Now simplify:
Add the constant of integration:
The answer is:
/ 3 \ / | cos (x)| | 3 / 3 \ 2*x*|-2*cos(x) - -------| | 2 3 14*sin(x) 2*sin (x) 2 | sin (x) | \ 3 / | x *cos (x) dx = C - --------- + --------- + x *|- ------- + sin(x)| - ------------------------- | 9 27 \ 3 / 3 /
3 3 2 2 22*sin (1) 14*cos (1) 5*cos (1)*sin(1) 4*sin (1)*cos(1) - ---------- + ---------- - ---------------- + ---------------- 27 9 9 3
=
3 3 2 2 22*sin (1) 14*cos (1) 5*cos (1)*sin(1) 4*sin (1)*cos(1) - ---------- + ---------- - ---------------- + ---------------- 27 9 9 3
Use the examples entering the upper and lower limits of integration.