The Basic Idea
Integration reverses differentiation. If you know the derivative of a trigonometric function, you can work backward to find its antiderivative.
The two most important identities to memorize are:
Notice the negative sign in the antiderivative of cosine. This is one of the most common places students make a mistake.
Because the derivative of is , the antiderivative of must be . Likewise, because the derivative of is , the antiderivative of is .
The Core Trigonometric Antiderivatives
For AP Calculus, you should know the following relationships automatically:
These formulas are simply the reverse of the corresponding derivative rules. For example, since:
we know that:
Remember that every indefinite integral includes a constant of integration .
Integrating Sine and Cosine
The easiest trigonometric integrals are those involving sine and cosine directly. Start by asking: what function has this expression as its derivative?
Evaluate .
Evaluate .
Constant Multiples
A constant multiplying a trigonometric function can simply be carried through the integration.
For example, if the integrand is , the coefficient remains attached to the antiderivative.
Evaluate .
When the Angle Is Not Just x
The chain rule has an integration counterpart. If the argument of sine or cosine is something like , you need to account for the derivative of the inside function.
Likewise:
Evaluate .
Integrating Secant and Cosecant
Two especially important trigonometric antiderivatives are:
These are worth memorizing because the derivatives of and produce exactly the expressions needed.
For example:
Therefore:
Evaluate .
Integrating Tangent and Cotangent
For tangent and cotangent, rewrite the functions using sine and cosine:
Then use the logarithmic antiderivatives:
Evaluate .
Definite Integrals
For a definite integral, evaluate the antiderivative at the upper and lower limits and subtract:
The constant of integration is not needed when evaluating a definite integral because it cancels when the upper and lower values are subtracted.
Evaluate .
Using Trigonometric Identities Before Integrating
Sometimes the integrand is not immediately recognizable. In those cases, a trigonometric identity can transform the expression into something that is easier to integrate.
One of the most useful identities is the Pythagorean identity:
For example, an expression involving can sometimes be rewritten using .
Evaluate .
A Fast Recognition Strategy
On an AP Calculus problem, do not immediately start manipulating the expression. First identify the derivative pattern.
- See cosine? Think .
- See sine? Think .
- See secant squared? Think .
- See cosecant squared? Think .
- See secant times tangent? Think .
- See cosecant times cotangent? Think .
- See tangent? Rewrite it as .
Common Mistakes
The derivative of cosine is negative sine. Therefore, the antiderivative of sine is negative cosine, not positive cosine.
For an integral such as , the answer is not simply . You must account for the derivative of the inside function.
Every indefinite integral represents a family of antiderivatives, so include unless you are evaluating a definite integral.
When integrating tangent, remember that , so the logarithmic antiderivative involves .
Practice Problems
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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