Question

Evaluate the integral by making the given substitution. (Use c for the constant of integration.) ∫(1 − 6t)⁶ dt, where u = 1 − 6t.

a. ∫u⁶ du
b. (1/6)u⁶ + c
c. (1/7)u⁷ + c
d. ∫(1 − u)⁶ du

Answer 1

To evaluate the integral, we can use the given substitution and follow the steps of substitution and integration.

Step-by-step explanation:

To evaluate the integral ∫(1 − 6t)⁶ dt using the given substitution u = 1 − 6t, we can follow these steps:

Differentiate u with respect to t: du/dt = -6

Rearrange the equation to solve for dt: dt = -du/6

Substitute the expressions for u and dt into the integral expression: ∫(1 − 6t)⁶ dt = ∫u⁶ (-du/6)

Simplify the expression: (1/6)∫u⁶ du

Integrate u⁶ with respect to u: (1/6)(u⁷/7) + C, where C is the constant of integration

Substitute back for u: (1/6)((1 − 6t)⁷/7) + C

So, the correct answer is (1/6)((1 − 6t)⁷/7) + C.