Question

Please help me solve part c. How do I rewrite the integral without using a trig function?

(b) Let I = the integral of [(x − 1)² + sin(x − 1) + (x − 1)³]dx from -1 to 3. Use the substitution u = x − 1 to rewrite I as an integral in the variable u.
(c) Then rewrite the integral you found in part (b) as an integral of only the function u² (your answer should not contain any trig functions or any powers of u except for u²)

Answer 1

To rewrite the integral without trig functions or powers of u except u², leverage previous problem results or use trigonometric identities to transform sin(x - 1) into a function of u².

Step-by-step explanation:

The correct answer is option to rewrite the integral without any trigonometric functions or any powers of u except for u². This is possible because the terms in the integrand that are not u² can be manipulated or removed based on the given problem constraints or previous results.

For the term involving sin(x - 1), trigonometric identities or known integrals may simplify the expression to only involve u². If any term like (x - 1)³ can't be rewritten as a function of u², it might be an indication that further information is required from the previous parts of the problem, or that a mistake has been made.

Without the full context or specific rules given in the problem, it's not possible to provide a definitive transformation.

Now, we want to rewrite the integral in terms of u² only. Notice that u³ = u * u². We can split the integral into two parts:

I = ∫u² du + ∫sin(u) du + ∫u * u² du

The first and third integrals are integrals of u² terms and don't contain any trig functions or any powers of u except for u², which is what we want.