Integration Formulas Cheat Sheet

This integration formulas cheat sheet collects the standard antiderivative rules students reach for 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 core skill is pattern matching: a sum or difference can be integrated term by term, while a product, quotient, or composition usually needs another method.

The Formulas And Their Symbols

In every rule, $C$ is the constant of integration and the variable of integration is $x$. The power rule and its exponent condition come first:

• 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 ($a$ is the base):

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

• Basic trig rules:

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

Linearity links most of these together:

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

It works for sums and differences but does not let you split a product into separate integrals.

Why These Rules Hold, And The Exception They Force

Each formula is a derivative read backwards, so differentiating the right side returns the integrand. For the power rule, differentiating $\frac{x^{n+1}}{n+1}$ gives $\frac{(n+1)x^n}{n+1} = x^n$. That same algebra exposes why $n = -1$ is excluded: there $x^n = x^{-1} = \frac{1}{x}$, and the antiderivative turns logarithmic:

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

Writing $\frac{x^0}{0}$ would be meaningless, which is exactly why this case is handled separately. Reading the formulas as reversed derivatives also hands you a check for every answer.

Worked Example, Step By Step

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:

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

Then check by differentiating:

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

Returning to the integrand is the fastest way to catch a sign error.

Try It Yourself

Work $\int \left(6x - 2\cos x + \frac{3}{1+x^2}\right)\,dx$ 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.

Calculation Pitfalls

1. Forgetting the constant of integration. Indefinite integrals 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, such as polynomials, basic trig functions, and simple exponentials. If it does not match a known form, stop before forcing a formula: products often call for integration by parts, and compositions often call for substitution.