integral 3e^(3x)sin(e^3x)dx
how about you try this one? what do you get?
hint: let u = e^3x and you have u sinu, use integration by parts
To find the integral of 3e^(3x)sin(e^3x)dx, we can use a technique called u-substitution. Let's go through the process step by step:
Step 1: Identify the appropriate substitution.
In this integral, let's choose u = e^3x. Taking the derivative of u with respect to x gives du/dx = 3e^3x. Notice that the expression 3e^3x is very similar to one of the terms in our integrand: 3e^(3x). This suggests that substituting u will likely simplify the integral.
Step 2: Calculate the differential du.
To find du, we can rearrange the equation from Step 1 and divide both sides by 3:
du = du/dx dx
du = 3e^3x dx
Step 3: Substitute the variables.
Now, substitute u = e^3x and du = 3e^3x dx into the integral:
∫ 3e^(3x)sin(e^3x) dx = ∫ sin(u) du
Step 4: Evaluate the new integral.
The integral of sin(u) is -cos(u). So, we can rewrite our integral as:
∫ sin(u) du = -cos(u)
Step 5: Convert back to the original variable.
To convert back to the original variable x, we can substitute u = e^3x back into the expression above:
- cos(u) = - cos(e^3x)
Step 6: Add the constant of integration.
Remember that when integrating, we need to add a constant of integration (C) to the result:
∫ 3e^(3x)sin(e^3x) dx = -cos(e^3x) + C
So, the final result is -cos(e^3x) + C, where C is the constant of integration.