Evaluate the indefinite integral.
e^(8x)sin(e^(8x))
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To evaluate the indefinite integral ∫e^(8x)sin(e^(8x)) dx, we can use the technique of integration by substitution.
Let's make the substitution u = e^(8x). Taking the derivative of both sides with respect to x, we get du/dx = 8e^(8x). Rearranging this equation, we have du = 8e^(8x) dx.
Substituting u and du into the integral, we have:
∫e^(8x)sin(e^(8x)) dx = ∫sin(u) (1/8) du
Now, we can easily integrate sin(u) with respect to u:
∫sin(u) du = -cos(u) + C
Therefore, the original integral becomes:
= ∫sin(u) (1/8) du = (1/8)(-cos(u)) + C = (-1/8)cos(u) + C
Finally, we substitute back in u = e^(8x) to get the final answer:
= (-1/8)cos(e^(8x)) + C
Therefore, the indefinite integral of e^(8x)sin(e^(8x)) is (-1/8)cos(e^(8x)) + C, where C represents the constant of integration.
To evaluate the indefinite integral of e^(8x)sin(e^(8x)), we can use integration by parts. Integration by parts is a technique that allows us to split up the integral of a product of functions.
The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's assign u and dv to the functions in our integral.
u = sin(e^(8x))
dv = e^(8x) dx
Next, we need to find du and v.
To find du, we take the derivative of u with respect to x.
du = cos(e^(8x)) * (8e^(8x)) dx
To find v, we integrate dv with respect to x.
v = ∫ e^(8x) dx
To integrate v, we can use substitution. Let's substitute u = 8x.
du = 8 dx
dx = du/8
v = ∫ e^u * (du/8)
= (1/8) ∫ e^u du
= (1/8) e^u + C, where C is the constant of integration.
Now that we have du and v, we can substitute these back into the integration by parts formula:
∫ e^(8x)sin(e^(8x)) dx
= uv - ∫ v du
= sin(e^(8x)) * ((1/8) e^(8x) + C) - ∫ ((1/8) e^(8x) + C) * (cos(e^(8x)) * (8e^(8x)) dx)
= (1/8) e^(16x) sin(e^(8x)) + C1 - (1/8) ∫ e^(16x) cos(e^(8x)) dx - C2
The remaining integral can be quite challenging to solve, as it involves the product of two exponential functions. There is no elementary function that can be expressed in terms of standard functions to evaluate this integral.
Therefore, the solution to the indefinite integral is:
∫ e^(8x)sin(e^(8x)) dx = (1/8) e^(16x) sin(e^(8x)) - (1/8) ∫ e^(16x) cos(e^(8x)) dx + C
where C = C1 - C2 is the constant of integration.
or, let u = e^8x
du = 8e^8x dx
The integrand then becomes
sin(u) du/8
If that's too difficult, you have some serious catching up to do.