Evaluate.
The integral of (e^5x)(sin6x)dx
Try integration by parts.
You will have to do it twice.
Be very careful, it is so easy to make mistakes in these kind of messy questions.
Here is an example very similar to yours
http://www.sosmath.com/calculus/integration/byparts/Example3/Example3.html
Notice the switch in substitution in the second round.
(Did you not have some easier questions as lead-ups for this type where one step was enough?)
unfortunately no. my professor is not very helpful and throws problems at us by skipping steps and providing half hearted explanations
Also thank you for the help! I now understand :)
To evaluate the integral ∫(e^5x)(sin6x)dx, we can use the technique of integration by parts. This method allows us to simplify the integrand by breaking it down into two functions, one of which we differentiate and the other we integrate.
The formula for integration by parts is:
∫u * v dx = u * ∫v dx - ∫u' * (∫v dx) dx
Where u is the function we choose to differentiate, and v is the function we choose to integrate.
Let's apply this formula to the given integral:
Let u = e^5x (the function we differentiate)
Let dv = sin6x dx (the function we integrate)
Taking the derivative of u, we get:
du = (d/dx)(e^5x) dx
= 5(e^5x) dx
Integrating dv, we get:
v = ∫sin6x dx = (-1/6)cos6x
Now, using the formula for integration by parts, we have:
∫(e^5x)(sin6x)dx = u * v - ∫u' * v dx
= e^5x * (-1/6)cos6x - ∫(5e^5x)(-1/6)cos6x dx
Simplifying this expression, we have:
∫(e^5x)(sin6x)dx = (-1/6) * e^5x * cos6x - (5/6) * ∫e^5x * cos6x dx
Now, we have a new integral to evaluate: ∫e^5x * cos6x dx. We can use integration by parts again on this integral.
Choose u = e^5x and dv = cos6x dx. Taking the derivatives and integrating, we have:
du = (d/dx)(e^5x) dx
= 5(e^5x) dx
v = ∫cos6x dx
= (1/6)sin6x
Applying integration by parts once again, we have:
∫e^5x * cos6x dx = u * v - ∫u' * v dx
= e^5x * (1/6)sin6x - ∫(5e^5x)(1/6)sin6x dx
Simplifying this expression, we have:
∫e^5x * cos6x dx = (1/6) * e^5x * sin6x - (5/6) * ∫e^5x * sin6x dx
Notice that the integral we obtained after applying integration by parts again is the same as the original integral we were trying to evaluate.
Thus, we can substitute this integral back into our original equation:
∫(e^5x)(sin6x)dx = (-1/6) * e^5x * cos6x - (5/6) * ∫e^5x * cos6x dx
Now, we have an equation with only one unknown, which is the original integral itself.
Rearranging the equation to isolate the integral, we get:
(1 + (5/6)) * ∫(e^5x)(sin6x)dx = (-1/6) * e^5x * cos6x
Multiplying both sides by (6/11), we have:
∫(e^5x)(sin6x)dx = (-1/11) * e^5x * cos6x
Therefore, the value of the integral ∫(e^5x)(sin6x)dx is (-1/11) * e^5x * cos6x + C, where C is the constant of integration.