   Chapter 13.2, Problem 40E

Chapter
Section
Textbook Problem

# Evaluate the integral.40. ∫ ( t e 2 t   i  +  t 1  -  t  j  +  1 1  -  t 2   k ) dt

To determine

To Evaluate: The integral (te2ti+t1tj+11t2k)dt.

Explanation

To find the integral of the vector function, integrate each component of the vector function.

Consider the integral (te2ti+t1tj+11t2k)dt as I to minimize the calculation.

Integrate each component of the vector function (te2ti+t1tj+11t2k) as follows.

I=[te2tdt]i+[(t1t)dt]j+[11t2dt]k (1)

Integrate x-component as follows.

te2tdt

Consider u=t and dv=e2tdt.

Write the formula to integrate te2tdt as follows.

udv=uvvdu (2)

Find v from the expression dv=e2tdt.

dv=e2tdt

Integrate on both sides of the expression.

dv=e2tdtv=e2t2+C

Here,

C is the constant.

Find du from u=t.

u=t

Differentiate on both sides of the expression.

du=dt

Integration of te2tdt:

Substitute t for u, e2t2 for v, and dt for du in equation (2),

te2tdt=te2t2(e2t2)dt=te2t2e2t4=12te2t14e2t+C

Rewrite the expression in equation (1) by substituting the value of x-component as follows.

I=(12te2t14e2t)i+[(t1t)dt]j+[11t2dt]k

Rewrite the expression as follows

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