1) Integrate Cos^n(x) dx
2) Integrate e^(ax)Sinbx dx
3) Integrate (5xCos3x) dx
I Will be happy to critique your thinking on these.
1) Derive a recursive relation.
2) Simplest by replacing sin(bx) by
Exp[i b x] and taking imaginary part at the end.
3)
First integrate sin(yx) and then differentiate the answer w.r.t. y.
1) To integrate cos^n(x) dx, we can derive a recursive relation.
First, let's consider the base case: n = 1. When n = 1, we have the integral of cos(x) dx, which is sin(x) + C, where C is the constant of integration.
Now, let's consider the case where n > 1. We can use integration by parts to derive a recursive relation. The integral of cos^n(x) dx can be written as:
∫ cos^n(x) dx = cos^(n-1)(x) * sin(x) + (n-1) * ∫ cos^(n-2)(x) * sin^2(x) dx
Now, we can use the double angle identity for sin^2(x):
sin^2(x) = (1 - cos(2x))/2
Substituting this back into the equation, we get:
∫ cos^n(x) dx = cos^(n-1)(x) * sin(x) + (n-1) * ∫ cos^(n-2)(x) * (1 - cos(2x))/2 dx
This equation gives us a recursive relation to solve for the integral of cos^n(x) dx. We solve the integral for decreasing values of n until we reach the base case where n = 1.
2) To integrate e^(ax)sin(bx) dx, a simpler approach is to replace sin(bx) by exp(i * bx) and take the imaginary part at the end.
First, let's consider replacing sin(bx) by exp(i * bx):
∫ e^(ax)sin(bx) dx = ∫ e^(ax) * (im[exp(i * bx)]) dx
Now, we can use Euler's formula to expand the imaginary part:
∫ e^(ax) * (im[cos(bx) + i * sin(bx)]) dx
Taking the imaginary part of the integral, we get:
im[∫ e^(ax) * (cos(bx) + i * sin(bx)) dx]
Now, we can split the integral into two parts, one for the real part (cos(bx)) and one for the imaginary part (sin(bx)):
im[∫ e^(ax) * cos(bx) dx + ∫ e^(ax) * i * sin(bx) dx]
Next, we integrate each part separately:
1/a * e^(ax) * cos(bx) - b/(a^2 + b^2) * e^(ax) * sin(bx) + C
So, the integral of e^(ax)sin(bx) dx is given by:
(1/a) * e^(ax) * cos(bx) - (b/(a^2 + b^2)) * e^(ax) * sin(bx) + C
3) To integrate (5xCos3x) dx, we can follow a two-step process:
First, integrate sin(yx), and then differentiate the solution with respect to y.
First, we integrate sin(yx) with respect to x:
∫ sin(yx) dx = (-1/y) * cos(yx) + C
Next, we differentiate the result with respect to y. Since y is a constant, the derivative of cos(yx) with respect to y is 0, so we only need to differentiate the coefficient of cos(yx) with respect to y:
(d/dy)(-1/y) = -1/y^2
So, the final result of integrating (5xCos3x) dx is:
(-1/3) * cos(3x) + C
Please note that C represents the constant of integration in all the solutions.