Question

Use the formula ∫u2sin(u)du=(2−u2)cos(u)+2usin(u)+C to eval ∫x106sin(x31)dx Be sure to put the arguments of any trigonometric functions in parentheses. Provide your answer below: ∫x106sin(x31)dx=

Answer 1

∫x¹⁰⁶sin(x³¹)dx = (2 - x³¹)cos(x³¹) + 2x³¹sin(x³¹) + C

Step-by-step explanation:

In order to evaluate the integral ∫x¹⁰⁶sin(x³¹)dx, we can employ the given formula ∫u²sin(u)du = (2 - u²)cos(u) + 2usin(u) + C, where 'u' is a variable. In our case, we let u = x³¹. Then, differentiate u with respect to x to obtain du/dx = 3x³⁰. Now, substitute these values into the formula:

∫x¹⁰⁶sin(x³¹)dx = ∫u²sin(u) * (1/3x³⁰) du

This becomes (1/3)∫u²sin(u)du. Now, apply the given formula, replacing 'u' with x³¹:

= (1/3) * [(2 - x³¹)cos(x³¹) + 2x³¹sin(x³¹)] + C

Simplify this expression to get the final answer:

= (2 - x³¹)cos(x³¹)/3 + 2x³¹sin(x³¹)/3 + C

So, ∫x¹⁰⁶sin(x³¹)dx = (2 - x³¹)cos(x³¹)/3 + 2x³¹sin(x³¹)/3 + C.

In this way, we have successfully applied the provided formula, making the substitution and carrying out the necessary calculations to determine the integral.