Let and be real constants and let be any differentiable function whose derivative is…

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

a\,(\neq 0)

and

b

be real constants and let

F

be any differentiable function whose derivative

F'

is continuous. Prove the following statement without evaluating any specific numerical integral: (a) Show that

\displaystyle\int F'(a x + b)\,dx = \dfrac{1}{a}\,F(a x + b)+C

, where

C

is the constant of integration. (b) Explain briefly why the factor

1/a

cannot be omitted even though it is a constant.