Question

Evaluate the integral. ∫₀¹​[(21t²−2)i+8t/t²+1j-t/√t²+1k]dt a.9i+4ln 2j+(1+√2​)k b.5i+4ln 2j+(1−√2​)k c.9i+4ln 2j+(1−√2​)k d.9i+4ln 2j+1−√2/2​​k​

Answer 1

To evaluate the given integral, perform separate integrations for each component (i, j, k) of the vector function. After integration, combine the results to obtain the final vector, which should match one of the provided options.

Step-by-step explanation:

The student is asking to evaluate the definite integral of a vector-valued function from 0 to 1. The vector function is given by [(21t²−2)i + (8t/t²+1)j − (t/√t²+1)k]. To solve this, we integrate each component of the vector function separately with respect to t over the interval from 0 to 1.

Steps for integration:

1. Integrate the i component: ∫₀¹ 21t² − 2 dt, which is a standard polynomial integral.
2. Integrate the j component: ∫₀¹ 8t/(t²+1) dt, which can be evaluated using a simple substitution or recognizing it as the derivative of a logarithm.
3. Integrate the k component: ∫₀¹ −t/√t²+1 dt, which requires a substitution to simplify the integral.
4. Finally, combine the results of the integrals to get the final vector.

After performing the integration for each component, we combine the results to get the final answer, which should match one of the given options (a, b, c, or d).