Final answer:
To evaluate the indefinite integral ∫x^3(5+x^4)^11dx using the change-of-variable technique, substitute u = 5 + x^4 and simplify the integral to ∫[(u^(11/4))/(4x^3)] du. Transform the limits of integration to u = 5 to u = 5005. Integrate (u^(11/4)) with respect to u to find the value of the integral.
Step-by-step explanation:
To evaluate the indefinite integral ∫x^3(5+x^4)^11dx using the change-of-variable technique, we can substitute u = 5 + x^4. Then, du = 4x^3dx. Rearranging the terms, we get dx = du/(4x^3). Substituting these values into the integral, we have ∫(u^(11/4))(du/(4x^3)). Now, we can simplify the integral by canceling out the x^3 term in the denominator. The integral becomes ∫[(u^(11/4))/(4x^3)] du.
Next, we need to transform the limits of integration. When x = 0, u = 5 + (0^4) = 5. When x = 10, u = 5 + (10^4) = 5005. So, the limits of integration become 5 to 5005.
Now we can proceed with integrating. The integral becomes (1/4)∫[(u^(11/4))/(x^3)] du. We can pull out the constant (1/4) and simplify the integral to (1/4) ∫(u^(11/4)) du.
Integrating u^(11/4) with respect to u gives us (1/4)(u^(15/4))/(15/4), which simplifies to (1/4)(4/15)(u^(15/4)).
Plugging back in the values of u and simplifying, we evaluate the integral as ((1/4)(4/15)(5005^(15/4))) - ((1/4)(4/15)(5^(15/4))).