How to Integrate on the TI-89: Indefinite, Definite and Exact Results

You want to know how to integrate on the TI-89 Titanium, and the answer either looks right but lacks a +C, or a definite integral comes back as a long unevaluated expression. Both have simple explanations once you know the two integration commands and when each one is exact.

Before you integrate: HOME, MODE and old variables

Work on the HOME screen. In MODE, set Angle to RADIAN for any integrand with sin, cos or tan, and leave Exact/Approx on AUTO so exact input gives exact output. Then clear stale variables: if x still holds a number from an earlier problem, ∫( integrates a constant. Press 2nd F1 for the Clean Up menu and pick NewProb.

Indefinite integrals: ∫(expression, variable)

The integral sign is the 2nd function of the 7 key. With two arguments, ∫(expression, variable) returns an antiderivative. It also sits in the F3 Calc menu as item 2, but 2nd 7 is faster. The full reference is on the ∫( page.

Type the integrand, a comma, the variable and the closing parenthesis, then press ENTER. Use the X key for x and the times key between factors, so the CAS does not read xe as one variable name.

The second row is integration by parts, and you do not set anything up for it: substitution, parts and partial fractions all happen inside the CAS. The answer is one antiderivative without the constant of integration, so write + C on paper yourself.

Definite integrals and integrals with limits

For a TI-89 definite integral, add the lower and the upper bound as the third and fourth arguments: ∫(expression, variable, lower, upper). For the area under x² from 0 to 1 you type ∫(x^2,x,0,1). The bounds can be numbers, π, other variables or ∞, which is ◆ CATALOG.

With exact bounds and an exact integrand, the result stays exact: fractions, roots, π and logarithms instead of decimals. That is what most free response questions want to see first. If the question asks for three decimal places, press ◆ ENTER instead of ENTER and the same input comes back as a decimal.

When the exact answer will not simplify

A common forum complaint: a definite integral in EXACT mode returns the integral itself, or a long expression with error functions, instead of a number. That happens when the integrand has no elementary antiderivative, or when the CAS cannot find one. You have three ways out.

• Press ◆ ENTER. The calculator evaluates the same input numerically.
• Use nInt( with the same four arguments. It always returns a decimal.
• Wrap the result in approx(, item 5 in the F2 Algebra menu.

nInt( is the right tool when you only need a number, for example to check a Riemann sum or a trapezoid estimate. It is also the safer choice for integrands that jump or oscillate, because it never tries to find a formula first. Type ∫(x^2,x,0,1) next to it and compare: the symbolic command gives the exact fraction, the numeric one the decimal.

Arc length, area between curves and double integrals

Arc length is an integral too, and it has its own command: arcLen( takes the same four arguments as a definite integral and builds the square root formula for you. Open F3 and scroll down to it.

The result shows why exact mode is worth keeping: you get a closed form with √10 that you can copy into a written answer, and ◆ ENTER turns it into a decimal when you need one.

For the area between two curves, integrate the top function minus the bottom function: ∫(f(x)-g(x),x,a,b). Find the intersection points a and b first with solve( from the F2 menu. If you defined both functions with Define, you can call them by name inside the integral.

A TI-89 double integral is two ∫( commands nested inside each other. The inner integral runs first, so write it with its own variable and bounds: ∫(∫(x·y,y,0,x),x,0,1) integrates over y from 0 to x, then over x from 0 to 1. The inner bounds may depend on the outer variable, the outer bounds must be numbers or constants. Triple integrals work the same way with one more level.

Common mistakes

• A stored value in x. The integral comes out as a multiple of x or as a number that makes no sense. Run NewProb, or use another variable.
• Degree mode with trig. In DEGREE mode, trig integrals come back with a conversion factor you did not expect. Switch to RADIAN.
• Missing times sign. Implicit products like 2x work, but xy or xe are read as variable names. Type x·y.
• Bounds in the wrong order. Swapping lower and upper flips the sign of the result.

In Calc89

Calc89 is a CAS graphing calculator intended for use with TI-89 Titanium, and the integral keys are the same: 2nd 7 for ∫(, ◆ CATALOG for ∞, ◆ ENTER for a decimal. Every row in the tables above comes from a verified example of the Calc89 CAS. If the integral is on a worksheet, scan it with the camera and Calc89 sets up the input for you. The AI keystroke tutor then shows the key sequence for the TI-89 Titanium step by step, so you can repeat it on the handheld. Calc89 is free to download, Pro features as an in-app purchase.

Frequently asked questions

Related pages

• /commands/integral/
• /commands/nint/
• /commands/arclen/
• /commands/approx/
• /guides/ti-89-ap-calculus/
• /blog/ti-89-derivatives/
• /blog/ti-89-taylor-series-sums/

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