   Chapter 13.2, Problem 42E

Chapter
Section
Textbook Problem

# Find r(t) if r'(t) = t i + et j + tet k and r(0) = i + j + k.

To determine

To find: The vector r(t) .

Explanation

Given data:

r(t)=ti+etj+tetk and r(0)=i+j+k .

Formula used:

Consider following integration formulae.

tndt=tn+1n+1+Cetdt=et+C

The tangent vector r(t)=ti+etj+tetk is the derivative of the vector r(t) .

To find the vector r(t) from the tangent vector r(t)=ti+etj+tetk , integrate each component of the vector r(t)=ti+etj+tetk .

Integrate each component of the vector function r(t)=ti+etj+tetk as follows.

r(t)dt=[(t)dt]i+[(et)dt]j+[(tet)dt]k

Rewrite the expression as follows.

r(t)=[(t)dt]i+[(et)dt]j+[(tet)dt]k (1)

Integration of z-component (tet)dt :

Consider u=t and dv=etdt .

Write the formula to integrate (tet)dt as follows.

udv=uvvdu (2)

Find v from the expression dv=etdt .

dv=etdt

Integrate on both sides of the expression.

dv=etdtv=et

Find du from u=t .

u=t

Differentiate on both sides of the expression.

du=dt

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

(tet)dt=tet(et)dt=tetet+C

Here,

C is the constant.

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

r(t)=[(t)dt]i+[(et)dt]j+[tetet+C]k

Compute the expression r(t)=[(t)dt]i+[(et)dt]j+[tetet+C]k by using the integral formulae as follows

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