Integration

Raise the power by one, divide by the new power, add a constant:

As with differentiation, rewrite roots and fractions as powers before you start: $\sqrt{x}=x^{1\mathrm{/}2}$, $\frac{3}{x^2}=3x^{-2}$. The negative-power cases are where signs go wrong: $\int 3x^{-2}dx=-3x^{-1}+c$ — raising $-2$ by one gives $-1$, and dividing by $-1$ flips the sign.

The chain rule in reverse: raise the power, divide by the new power, and divide by the bracket’s coefficient:

Example: $\int (3x-2{)}^{5}dx=\frac{(3x-2{)}^6}{18}+c$. The divide-by-$a$ is the single most forgotten step in P1 integration — checking by differentiating your answer takes five seconds and catches it every time.

When a question gives $\frac{dy}{dx}$ and a point on the curve, the constant is recoverable — and required:

1. Integrate, writing $+c$.
2. Substitute the given point into your expression.
3. Solve for $c$ and state the full equation of the curve.

“Find the equation of the curve” questions are marked on exactly this chain; an answer with no $c$ (or with $c$ never evaluated) caps the marks immediately.

A definite integral has limits and produces a number — substitute the top limit, substitute the bottom limit, subtract:

Write the square-bracket stage explicitly — it carries the method mark — and keep the subtraction in brackets: most sign errors on definite integrals happen when the lower limit’s value is negative and the brackets were skipped.

The area between a curve and the $x$-axis from $a$ to $b$ is ${\int }_a^{b}ydx$. Two refinements matter in P1:

• Region below the axis: the integral comes out negative; the area is its magnitude. If the region crosses the axis, split at the root and add the pieces’ absolute values.
• Between two curves: ${\int }_a^b(y_{\text{top}}-y_{\text{bottom}})dx$, with $a$ and $b$ the intersection points — found by solving the curves equal, with working shown.

1. 1.Sign errors on negative powers

   $\int x^{-3}dx=\frac{x^{-2}}{-2}+c=-\frac{1}{2x^2}+c$. Raise the power towards zero and let the division carry the sign — then differentiate back to check.

2. 2.Forgetting to divide by the bracket's coefficient

   $\int (3x-2{)}^{5}dx$ needs division by both 6 and 3. If the inside of the bracket has a coefficient other than 1, your answer owes a division by it.

3. 3.Dropping + c in indefinite integrals

   Every indefinite integral ends $+c$ — and in “find the equation of the curve” questions the whole point is evaluating it. Write it the moment you integrate.

4. 4.Treating below-axis area as negative (or ignoring the crossing)

   Areas are positive. If the curve crosses the axis inside your limits, one integral straight across cancels area against itself — split at the root, take magnitudes, add.

5. 5.Skipping the bracketed subtraction stage in definite integrals

   Write $[F(x){]}_a^b=F(b)-F(a)$ with both values in brackets before simplifying. The stage is a method mark, and the brackets are what stop $-(-2)$ becoming $-2$.