Evaluate the integral by using substitution. Use an upper-case "C" for the constant of integration.
integral x^5((x^6-2)^10)dx
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let z = x^6-2
then dz = 6 x^5 dx
I think you can take it from there :)
To evaluate the integral ∫ x^5((x^6-2)^10) dx using substitution, we can let u = x^6 - 2. Then, differential du will be equal to 6x^5 dx.
To solve for dx, divide both sides of the equation by 6x^5, which gives us dx = du/(6x^5).
Now, we can substitute u and dx in terms of du into the original integral:
∫ x^5((x^6-2)^10) dx = ∫ (x^5 * u^10) * (du/(6x^5))
Canceling out x^5 terms:
= (1/6) ∫ u^10 du
Now, we can simplify the integral:
= (1/6) * (u^11/11) + C
Replacing u with the original expression:
= (1/6) * ((x^6-2)^11/11) + C
Therefore, the final answer is (1/6) * ((x^6-2)^11/11) + C, where C represents the constant of integration.
To evaluate the integral ∫x^5((x^6-2)^10)dx using substitution, we can let u = x^6-2. This choice of substitution is motivated by the fact that the derivative of x^6-2 is 6x^5, which appears as a factor in the integral.
First, let's find du, the differential of u, in terms of dx:
du = d(x^6-2)
du = 6x^5dx
Now, we can express the original integral in terms of u:
∫x^5((x^6-2)^10)dx = ∫((x^6-2)^10)(x^5)dx
= ∫u^10 (1/6)du
= (1/6) ∫u^10 du
Integrating u^10, we get:
(1/6) * (1/11)u^11 + C
Replacing u with x^6-2, we have:
(1/6) * (1/11)(x^6-2)^11 + C
So, the final answer is:
(1/66)(x^6-2)^11 + C