This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
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For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of is measured as .
The function, denoted , is defined as the composite of the cube function and the secant function (which in turn is the composite of the reciprocal function and the cosine function). Explicitly, it is given as:
We can differentiate this function as a composite function, using the chain rule for differentiation. We get:
Alternatively, we could consider it as , and hence as a composite of and . With that interpretation, the chain rule for differentiation yields:
The two answers can be verified to be equivalent.
We use the recursive version of integration by parts.
We rewrite and perform integration by parts, taking as the part to integrate. We use that an antiderivative of is whereas the derivative of is :
We now use the fact that , or more explicitly, , to rewrite this as:
We now use the integration of the secant function to simplify this as:
We can choose an antiderivative of so that the above equality (between the left-most and right-most expression) holds without any additive constant adjustment, and we get:
We rearrange and obtain:
Dividing by 2, we get:
The general antiderivative expression is thus: