Question

Evaluate the indefinite integral as a power series. ∫ (t - t⁵) dt

a) t²/2 - t⁶/6 + C
b) t²/2 + t⁶/6 + C
c) t³/3 - t⁷/7 + C
d) t³/3 + t⁷/7 + C

Answer 1

The indefinite integral of (t - t^5) dt is t^2/2 - t^6/6 + C, found by integrating each term individually and including the constant of integration.

Step-by-step explanation:

The correct choice for evaluating the indefinite integral of (t - t5) dt is t2/2 - t6/6 + C.

To evaluate the indefinite integral, integrate each term separately by increasing the exponent by one and dividing by the new exponent. For the first term, integrating t gives t2 divided by 2. For the second term, integrating -t5 yields -t6 divided by 6. \

Don't forget to add the constant of integration, C, at the end. The negative sign remains with the t6 term because integration is a linear operation, which preserves the sign of the term being integrated.To evaluate the indefinite integral of (t - t⁵), we can distribute the integral sign across the terms: ∫ t dt - ∫ t⁵ dt.

The integral of t with respect to t is (t²/2) + C.The integral of t⁵ with respect to t is (t⁶/6) + C.Therefore, the indefinite integral of (t - t⁵) is (t²/2) - (t⁶/6) + C, which simplifies to t³/3 + t⁷/7 + C.