First make a substitution and then use integration by parts to evaluate.
The integral of (x^9)(cos(x^5))dx
What do you substitute first? I do not understand what they are asking for. Please help. thanks!
let integral (x^9)(cos(x^5))dx
= integral (1/5)x^5( (5x^4)cos(x^5))dx
let dv= (1/5)x^4cos(x^5)dx , let u = (1/5)x^5
v = sin(x^5) , du = x^4 dx
so integral of (x^9)(cos(x^5))dx
= uv - integral [vdu
= (1/5)x^5(sin(x^5) - integral [ x^4(sin(x^5))dx
= (1/5)x^5(sin(x^5) - (1/5)(-cos(x^5)
= (1/5)[ (x^5(sin(x^5)) + cos(x^5) ]
To evaluate the integral of (x^9)(cos(x^5))dx using integration by parts, you first need to identify the substitution.
A good choice for substitution would be letting u = x^5, so du = 5x^4 dx.
Now, rewrite the integral using this substitution:
∫ (x^9)(cos(x^5))dx = ∫ (1/5)(x^4)(5x^4)(cos(u))dx
∫ (x^9)(cos(x^5))dx = (1/5) ∫ (x^4)(cos(u))(5x^4)dx
Next, integrate the new integral ∫ (x^4)(cos(u))(5x^4)dx using integration by parts.
u = (x^4) and dv = (cos(u))(5x^4)dx
Differentiating u, we get du = 4x^3 dx
Integrating dv, we get v = (1/5) sin(u)
Now we can use the integration by parts formula:
∫ u dv = uv - ∫ v du
Plugging in the values we have, we get:
∫ (x^4)(cos(u))(5x^4)dx = (x^4) ((1/5) sin(u)) - ∫ ((1/5) sin(u))(4x^3)dx
Simplifying further:
∫ (x^4)(cos(u))(5x^4)dx = (1/5) (x^4) sin(u) - (4/5) ∫ (sin(u))(x^3)dx
At this point, we can go back to the original variable x by substituting back u = x^5:
∫ (x^4)(cos(u))(5x^4)dx = (1/5) (x^4) sin(x^5) - (4/5) ∫ (sin(x^5))(x^3)dx
Now you can evaluate the remaining integral by simplifying and using other integration techniques if needed.