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Chapter 4: Q. 52 (page 400)

Find a function fthat has the given derivative f'and value f(c). Find an antiderivative of f'by hand, if possible; if it is not possible to antidifferentiation by hand, use the Second Fundamental Theorem of Calculus to write down an antiderivative.

f鈥�(x)=2sin鈦�(蟺x),f(2)=4

Short Answer

Expert verified

Ans: The function is,f(x)=12(ln鈦�|2x鈭�1|)+3

Step by step solution

01

Step 1. Given information.

given,

f鈥�(x)=2sin鈦�(蟺x),f(2)=4

02

Step 2. The objective is to find a function f meeting the above values.

So,

f(x)=鈭�2sin鈦�(蟺x)dx=鈭�1ydy2=12(ln鈦�|y|)+c=12(ln鈦�|2x鈭�1|)+c

The function is, 12(ln鈦�|2x鈭�1|)+c

03

Step 3. Finding the value of c,

f(1)=312(ln鈦�|2(1)鈭�1|)+c=312ln鈦�1+c=3c=3

Therefore, the function is f(x)=12(ln鈦�|2x鈭�1|)+3.

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