When an antiderivative (F(x)= f(x) ,dx ) must satisfy the condition (F(x_0)=y_0 )

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

Problem

When an antiderivative

F(x)=\displaystyle\int f(x)\,dx

must satisfy the condition

F(x_0)=y_0

, which statement correctly explains how the constant

C

is fixed? (a) Substituting

x_0

into

F(x)

makes

C

equal to

y_0

directly. (b)

C

remains arbitrary because a single point condition cannot determine it. (c) By substituting

x_0

and

y_0

into

F(x)+C

, we obtain a linear equation in

C

whose solution uniquely fixes its value. (d)

C

becomes zero whenever

x_0=0

since the integral from

0

always starts at the origin.