e^3xsen2xdx
1 / | | 3 | e *x*sin(2*x)*1 dx | / 0
Integral(E^3*x*sin(2*x)*1, (x, 0, 1))
The integral of a constant times a function is the constant times the integral of the function:
There are multiple ways to do this integral.
Use integration by parts:
Let and let .
Then .
To find :
There are multiple ways to do this integral.
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 sine is negative cosine:
So, the result is:
Now substitute back in:
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:
Now evaluate the sub-integral.
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 integral of a constant times a function is the constant times the integral of the function:
Use integration by parts:
Let and let .
Then .
To find :
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:
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:
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 cosine is sine:
So, the result is:
Now substitute back in:
So, the result is:
The integral of a constant is the constant times the variable of integration:
The result is:
So, the result is:
So, the result is:
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | | 3 /sin(2*x) x*cos(2*x)\ 3 | e *x*sin(2*x)*1 dx = C + |-------- - ----------|*e | \ 4 2 / /
/ cos(2) sin(2)\ 3 |- ------ + ------|*e \ 2 4 /
=
/ cos(2) sin(2)\ 3 |- ------ + ------|*e \ 2 4 /
Use the examples entering the upper and lower limits of integration.