evaluate the indefinite integral:
x^13(cos(x^7))dx
I've tried using integration by parts but I haven't ssem to be getting the right answer. I've used u=x^13 and dv=cos(x^7) and vice versa...
much help is appreciated
Let x^7 = u
7 x^6 dx = du
The integral becomes.
(integral of)(1/7) u cos u du
Perform that integration by parts, which is a lot easier, and then substitute back x^7 for u.
To evaluate the indefinite integral ∫x^13cos(x^7)dx, you need to apply the technique of integration by substitution rather than integration by parts.
Let's start by making a substitution with u = x^7. To find du, differentiate both sides with respect to x: du/dx = 7x^(7-1) = 7x^6.
Rearrange this equation to solve for dx: dx = du / (7x^6).
Now substitute these expressions into the integral:
∫x^13cos(x^7)dx = ∫(x^13)(cos(u))*(du/(7x^6))
= (1/7) ∫x^7*x^6*cos(u)du
= (1/7) ∫x^13*cos(u)du.
Now, the integral with respect to u can be evaluated since the cos(u) term does not depend on x:
(1/7) ∫x^13*cos(u)du = (1/7) * ∫x^13 * cos(u) du
= (1/7) * sin(u) + C, where C is the constant of integration.
Finally, substitute the original variable back in:
(1/7) * sin(u) + C = (1/7) * sin(x^7) + C.
Thus, the solution to the original indefinite integral is:
∫x^13cos(x^7)dx = (1/7) * sin(x^7) + C, where C is the constant of integration.