   Chapter 4.9, Problem 45E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find f.f″(x) = ex − 2 sin x, f(0) = 3, f(π/2) = 0

To determine

To find: The general antiderivative for the function f(x)=ex2sinx,f(0)=3,f(π2)=0 .

Explanation

Given Data:

Write the given function as follows.

f(x)=ex2sinx,f(0)=3,f(π2)=0

Formula used:

The antiderivative function for the function ex is ex+C .

Here, C is the constant.

The antiderivative function for the function sinx is cosx+C .

The antiderivative function for the function cosx is sinx+C .

Calculation of f(x) :

Rewrite the function f(x)=ex2sinx as follows.

f(x)=ex2sinx (1)

From the antiderivative function formula, the antiderivative for the function in equation (1) is written as follows.

f(x)=ex2(cosx)+C=ex+2cosx+C

Calculation of f(x) :

Rewrite the function f(x)=ex+2cosx+C as follows.

f(x)=ex+2cosx+Cx0 (2)

From the antiderivative function formula, the antiderivative for the function in equation (2) is written as follows.

f(x)=ex+2(sinx)+C(x0+10+1)+D

f(x)=ex+2sinx+Cx+D (3)

As f(0)=3 , substitute 0 for x in equation (3),

f(0)=e0+2sin(0)+C(0)+D=1+0+0+D=1+D

Rewrite the expression as follows

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