Integration Basics

Integration reverses differentiation. An indefinite integral finds a function whose derivative is the one you were given; a definite integral turns that into a number between two bounds.

Basic power rule
Constant multiple rule
Sum and difference rule
Trigonometric integrals
Exponential and logarithmic integrals
Definite integrals

How to start

Split the integrand into terms (sum and difference rule).
Pull out constant factors (constant multiple rule).
Match each piece to a basic form: power, trigonometric, exponential or reciprocal.
Differentiate your answer to check it. Add C for an indefinite integral.

Basic power rule

Use it when the variable is raised to a constant power other than −1. This reverses the derivative power rule: add one to the exponent, then divide by the new exponent.

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

Example. Integrate x^3.

Raise the exponent to 4 and divide by 4: \dfrac{x^4}{4}+C. Check: the derivative of \tfrac{x^4}{4} is x^3.

Watch out: the constant C is needed because many functions share the same derivative. For x^{-1}=\tfrac1x use the logarithm rule instead.

Try it in the calculator

Constant multiple rule

Use it when a constant multiplies the function. Move the constant outside, integrate what is left, and keep the constant in front.

\int c\,f(x)\,dx=c\int f(x)\,dx

Example. Integrate 5x^2.

5\int x^2\,dx=5\cdot\dfrac{x^3}{3}, so the answer is \dfrac{5}{3}x^3+C.

Watch out: only constants can be moved outside. You cannot pull a variable out of an integral.

Try it in the calculator

Sum and difference rule

Use it when the integrand is made of several terms added or subtracted. Integrate each term on its own, then combine. One constant C at the end is enough.

\int\bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx\pm\int g(x)\,dx

Example. Integrate x^3+2x.

\int x^3\,dx=\tfrac{x^4}{4} and \int 2x\,dx=x^2. The answer is \dfrac{x^4}{4}+x^2+C.

Watch out: there is no matching rule for products or quotients. Expand or rewrite them first, or use substitution or integration by parts.

Try it in the calculator

Trigonometric integrals

Use it when the integrand is a basic sine, cosine or similar function. Read the derivative table backwards.

\int\sin x\,dx=-\cos x+C\qquad\int\cos x\,dx=\sin x+C\qquad\int\sec^2x\,dx=\tan x+C

Example. Integrate 3\sin x-2\cos x.

Term by term: -3\cos x-2\sin x. The answer is -3\cos x-2\sin x+C.

Watch out: the integral of sine has a minus sign. Differentiate your answer to check it.

Try it in the calculator

Exponential and logarithmic integrals

Use it when the integrand is e to a power, or the reciprocal \tfrac1x. The reciprocal is the missing case of the power rule: it integrates to a logarithm. A coefficient k inside the exponent gets divided out.

\int e^{x}dx=e^{x}+C\qquad\int e^{kx}dx=\frac{e^{kx}}{k}+C\qquad\int\frac{1}{x}dx=\ln|x|+C

Example. Integrate e^{2x}.

The inside 2x has derivative 2, so divide by 2: \dfrac{e^{2x}}{2}+C.

Watch out: the integral of \ln x is not simple. It needs integration by parts, which the calculator shows step by step.

Try it in the calculator

Definite integrals

Use it when you want a number, such as the net area between the curve and the x-axis from a to b. Find an antiderivative F, then evaluate it at the top bound and subtract its value at the bottom bound. The constant C cancels, so you leave it out.

\int_a^b f(x)\,dx=F(b)-F(a)

Example. Find \int_0^3 x^2\,dx.

Antiderivative: F(x)=\tfrac{x^3}{3}.
Evaluate: F(3)=9 and F(0)=0.
Subtract: 9-0=9.

The value is 9.

Watch out: area below the x-axis counts as negative, and swapping the bounds flips the sign. In the calculator the bounds must be finite numbers or constants such as pi.

Try it in the calculator

Where integrals are used

Integrals add up small pieces: area under a curve, distance travelled from speed, total change from a rate. Products and compositions often need substitution or integration by parts, which the calculator names in its steps.

Open the interactive tools →