Integration Formulas Cheat Sheet

This integration formulas cheat sheet gives the standard antiderivative rules students use first in calculus. Use it when the integrand already matches a known pattern such as a power, $1/x$, an exponential, or a basic trig function.

The main job is pattern matching. If the expression is a sum or difference, you can usually integrate term by term. If it is a product, quotient, or composition, you may need another method instead.

Main Integration Formulas

• Power rule:

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

• Logarithm case:

$$\int \frac{1}{x}\,dx = \ln|x| + C$$

• Exponential rules:

$$\int e^x\,dx = e^x + C$$ $$\int a^x\,dx = \frac{a^x}{\ln(a)} + C, \qquad a > 0,\ a \ne 1$$

• Basic trig rules:

$$\int \sin x\,dx = -\cos x + C$$ $$\int \cos x\,dx = \sin x + C$$ $$\int \sec^2 x\,dx = \tan x + C$$ $$\int \frac{1}{1+x^2}\,dx = \arctan x + C$$

One rule connects most of these examples: linearity.

$$\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$

This works for sums and differences. It does not mean you can split a product into separate integrals.

The Exception Most Students Miss

The power rule does not work when $n = -1$. In that case, $x^n = x^{-1} = \frac{1}{x}$, and the antiderivative is logarithmic:

$$\int \frac{1}{x}\,dx = \ln|x| + C$$

Writing $\frac{x^0}{0}$ would be meaningless, which is why this case has to be handled separately.

Worked Example Using Several Integration Formulas

Find

$$\int \left(3x^2 - 4\sin x + 5e^x\right)\,dx$$

Each term matches a standard formula, so use linearity and integrate one term at a time:

$$\int 3x^2\,dx = x^3$$ $$\int -4\sin x\,dx = 4\cos x$$ $$\int 5e^x\,dx = 5e^x$$

Add the results and include the constant of integration:

$$\int \left(3x^2 - 4\sin x + 5e^x\right)\,dx = x^3 + 4\cos x + 5e^x + C$$

Check by differentiating:

$$\frac{d}{dx}\left(x^3 + 4\cos x + 5e^x + C\right) = 3x^2 - 4\sin x + 5e^x$$

That last step is the fastest way to catch a sign error.

Common Mistakes With Integration Formulas

1. Forgetting the constant of integration. For indefinite integrals, the answer should include $+C$.
2. Using the power rule when $n=-1$. $\int x^{-1}\,dx$ is not a power-rule case; it is $\ln|x| + C$.
3. Splitting a product as if integrals distribute over multiplication. In general, $\int f(x)g(x)\,dx \ne \left(\int f(x)\,dx\right)\left(\int g(x)\,dx\right)$.
4. Copying derivative formulas without reversing them carefully. For example, $\int \sin x\,dx$ is $-\cos x + C$, not $\cos x + C$.

When To Use an Integration Formula

Use a direct integration formula when the integrand already matches a standard pattern after simple algebra. Typical examples are polynomials, basic trig functions, and simple exponentials.

If the integrand does not match a known form, stop before forcing a formula. Products often call for integration by parts, and compositions often call for substitution.

Try a Similar Problem

Try $\int \left(6x - 2\cos x + \frac{3}{1+x^2}\right)\,dx$ on your own. If every term matches a standard formula and your final answer differentiates back to the original integrand, you are using the cheat sheet the right way.