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Chapter 4: Q. 54 (page 404)

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that fand gare continuous functions and thatk is any real number.
鈭�f'(g(x))g'(x)dx=_____

Short Answer

Expert verified

鈭�f'(g(x))g'(x)dx=f'(g(x))+C

Step by step solution

01

Step 1. Given Information

鈭�f'(g(x))g'(x)dx=_____

02

Step 2. Solving the integral

Let g(x)=t
Differentiate with respect to 't'
ddtg(x)=dtdt

g'(x)dx=dt
Putting the value鈭�f'(g(x))g'(x)dx=鈭�f'(t)dt

03

Step 3. Using the antiderivative

As 鈭�f(x)dx=F(x)+Cwhere Fis the antiderivative of f

Here antiderivative of f'(x)=f(x)
鈭�f'(g(x))g'(x)dx=f(t)+C
Putting t=g(x)
鈭�f'(g(x))g'(x)dx=f(g(x))+C

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