How to Solve Integrals Step by Step

Integration is differentiation in reverse. If a derivative tells you the slope of a curve, an integral tells you the area under it — or the original function you started from. Most integrals students meet are solved with one rule and a little care.

The power rule (the one rule you need most)

For any power of $x$:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$$

The $+C$ matters: it stands for the constant that differentiation erases. An indefinite integral always carries $+C$; a definite integral, with limits, does not — the constant cancels out.

Worked example 1: indefinite integral

$$\int 2xdx$$

Apply the power rule to $2x$: raise the power by one and divide by the new power.

$$\int 2xdx=x^2+C$$

Check by differentiating: the derivative of $x^2+C$ is $2x$. It matches, so the answer is right. Differentiating your answer is the fastest way to check any integral.

Worked example 2: definite integral

$${\int }_0^{1}2xdx$$

First find the antiderivative, $x^2$, then evaluate it at the limits and subtract:

$$[x^2{]}_0^1=1^2-0^2=1$$

So the area under $y=2x$ from $0$ to $1$ is $1$. Notice the order: upper limit minus lower limit. Reversing them flips the sign — a very common slip.

When the power rule is not enough

• Substitution: when the integrand contains a function inside another function, substitute the inner part (for example, $\int 2x\cdot e^{x^2}dx$ with $u=x^2$).
• Integration by parts: for products like $\int x{e}^{x}dx$.
• Partial fractions: for complicated rational functions.

The derivative rules cheat sheet helps here too: every differentiation rule read backwards is an integration rule. And if limits are involved, understanding limits covers the foundations.

Common mistakes

• Forgetting $+C$ on indefinite integrals.
• Subtracting the limits the wrong way round in definite integrals.
• Applying the power rule when $n=-1$. $\int \frac{1}{x}dx=\ln |x|+C$, not a power-rule result.
• Skipping the check. Differentiate your answer — if it does not give back the integrand, something went wrong.

Practice it

Want to check your work? Try the Integral Calculator — every answer is verified step by step. Then try similar problems on the practice page until the power rule feels automatic.