Forgetting the constant of integration

Integration reverses differentiation, and differentiation destroys constants: the derivative of $x^{2}+7$ and of $x^{2}-3$ is the same $2x$. So when you integrate $2x$ you cannot know which constant was there, you must write $+c$ to stand for every possibility.

The tempting mistake in Add Math is to leave it off, because the power-rule part feels like the whole answer and the $+c$ looks like decoration. On a plain "integrate this" question it may only cost a mark.

But the moment the question gives a point on the curve, or a value the function must take, the $+c$ becomes the unknown you are meant to solve for. Drop it and there is nothing left to substitute into, so the rest of the question collapses.

Integrate $6x^{2}-4x+1$. Applying the power rule term by term but stopping there gives:

An indefinite integral is a family of curves, not one curve, so the constant must be there:

Check by differentiating the right-hand side: $\frac{d}{dx}(2x^{3})=6x^{2}$, $\frac{d}{dx}(-2x^{2})=-4x$, $\frac{d}{dx}(x)=1$, and $\frac{d}{dx}(c)=0$. The $c$ vanishes on the way back, which is precisely why it has to be written on the way forward.

1. As you integrate, write $+c$ on the same line, before you do anything else, treat it as part of the integral sign, not an afterthought.
2. Ask whether the question is indefinite (an integral with no limits) or definite (with limits). Only indefinite integrals carry $+c$; in a definite integral it cancels out.
3. If a point or condition is given, substitute it into your $F(x)+c$ to form an equation in $c$.
4. Solve that equation for $c$, then write the final function with the specific value in place.
5. Verify by differentiating your answer back and checking it also passes through the given point.

Our teachers build the $+c$ into your hand: it goes down at the same moment the integral is written, not after. In a one-to-one lesson we practise the tell-tale phrase "passes through" or "when $x=\ldots,\ y=\ldots$", which signals that the constant is the whole point of the question.

Because the paper awards method marks, keeping the constant and forming the equation in $c$ earns credit step by step even if the arithmetic wobbles. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To lock in the habit, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.