Common exponential and trigonometric antiderivatives
Recall three fundamental antiderivatives.
Common exponential and trigonometric antiderivatives is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why common exponential and trigonometric antiderivatives works
These three are the matching derivative rules read backwards. Since e to the x repeats itself under differentiation it also repeats itself under integration, and since cosine is what sine differentiates to, cosine integrates back into sine. Sine needs a negative cosine because cosine differentiates into a negative sine.
What each symbol means
$C$ is an arbitrary constant.
Common exponential and trigonometric antiderivatives: when it holds
Angles are in radians; include an inner-derivative adjustment for composite inputs.
When it stops applying
All three assume a bare variable inside the function. The antiderivative of the cosine of 3x is not the sine of 3x, because differentiating that would leave an extra 3, so the result has to be divided by 3.
Common exponential and trigonometric antiderivatives: a worked example
$\int\cos(3x)dx=\frac13\sin(3x)+C$.
The mistake to avoid
What people do: Writing the antiderivative of sine as cosine with no negative sign.
Why it goes wrong: Differentiating cosine gives back a negative sine, not a sine, so the proposed answer fails its own check.
Do this instead: Differentiate any antiderivative you write down. The check takes seconds and catches every sign slip of this kind.
Common exponential and trigonometric antiderivatives: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $C$ is an arbitrary constant.
- Check the conditions before substituting. Angles are in radians; include an inner-derivative adjustment for composite inputs.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Integration
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where common exponential and trigonometric antiderivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about common exponential and trigonometric antiderivatives
Why does sine pick up a negative sign but cosine does not?
Because cosine differentiates into a negative sine. To land on a positive sine you must integrate to a negative cosine, and the sign is inherited from that step.
What is the antiderivative of e to the 5x?
It is one fifth of e to the 5x plus a constant, since differentiating brings a 5 down in front and the one fifth cancels it.
Do these work for negative inputs?
Yes, all three. The exponential, the sine, and the cosine are defined and continuous for every real number, so no restriction on the input appears.
Does the tangent belong in this group?
No. It needs a substitution and comes out as the negative natural logarithm of the absolute value of cosine, which is a logarithm rather than another trig function.