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Calculus IChapter 7: ExtrasSection 7-9

Constant of Integration — Question 6

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Question 6

Let ff be continuous on an interval II, and fix one antiderivative FF. Write the general antiderivative as y(x)=F(x)+C.y(x)=F(x)+C. Show that the condition y(x0)=y0,x0∈I,y(x_0)=y_0,\qquad x_0\in I, uniquely determines CC.

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Question 6 - Solution

Fix an antiderivative FF of ff on II. Every other antiderivative has the form y(x)=F(x)+Cy(x)=F(x)+C.

Imposing y(x0)=y0y(x_0)=y_0 yields F(x0)+C=y0F(x_0)+C=y_0, so

C=y0−F(x0).\boxed{C=y_0-F(x_0).}

This value is unique because FF, x0x_0, and y0y_0 are fixed.

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