Integrating Sums Of Basic Trigonometric Functions

Why Integrate a Sum One Term at a Time?

Many trigonometric expressions are sums of familiar terms. For instance, an integrand might contain sine, cosine, and secant-squared terms all at once. If each term has a known antiderivative, the sum does not require a new formula: integrate each term separately, keeping its coefficient and sign.

This method is an instance of linearity. It is useful because it turns one expression into several simpler tasks. The main care needed is to use the right basic formula for each term, and to work on an interval where all the terms are defined.

Linearity for Finite Sums

In symbols, the rule is:

This notation describes families of antiderivatives: the final result has one arbitrary constant, conventionally written \(+C\). The interval matters. For trigonometric functions with poles, choose an interval on which every term in the integrand is defined. In particular, do not treat a formula as giving an antiderivative across a point where the integrand is undefined.

Basic Trigonometric Antiderivatives

The formulas below are used term by term. Each formula applies on any interval where the functions involved are defined. The absolute values in the logarithms are part of the formulas; they allow the logarithm to be used when its argument is negative, as long as that argument is nonzero.

These are the standard basic antiderivatives. They can be checked by differentiation: for example, \(\frac{d}{dx}(-\cos x)=\sin x\), while \(\frac{d}{dx}(\tan x)=\sec^2 x\). The earlier material on checking trigonometric antiderivatives by differentiation gives the same central habit: verify a proposed result by differentiating it.

Why the Term-by-Term Rule Works

The rule follows directly from the derivative rules for sums and constant multiples. Suppose \(F_j'(x)=f_j(x)\) on an interval \(I\), for each \(j\) from \(1\) to \(n\). Define \(F(x)=a_1F_1(x)+\cdots+a_nF_n(x)\). Differentiating gives:

Thus \(F\) has exactly the required derivative, which proves it is an antiderivative of the sum. This argument explains why the method works rather than merely offering a procedure: the derivative of the proposed sum reproduces the original sum term by term.

On an interval, any two antiderivatives of the same function differ by a constant. Indeed, if \(F'=G'\), then \((F-G)'=0\). By the Mean Value Theorem, a function with derivative zero throughout an interval has the same value at any two points of that interval. Therefore \(F-G\) is constant there. This is why the general antiderivative is written with a single \(+C\).

A Reliable Procedure

Worked Examples

Common Pitfall: Losing a Sign

A frequent error is to forget that \(\int\csc^2 x\,dx=-\cot x+C\). For example, the calculation

is wrong: differentiating \(5\cot x\) gives \(-5\csc^2 x\), not \(5\csc^2 x\). The correction is:

Another bookkeeping error is to attach a coefficient to the wrong term when rewriting a sum. Keeping the original terms in order until each has been integrated makes this less likely. Differentiating the completed expression is the final safeguard.

Why This Method Matters

Term-by-term integration is one of the basic ways to manage a longer expression without changing the underlying problem. Once each term is recognized, the integral is assembled using linearity. The method does not say that every trigonometric expression can be integrated by simply splitting it into a sum: it applies when the integrand is a finite sum of terms whose antiderivatives are known, with constant coefficients. Products and more complicated compositions may require other methods.

Also, the constants from individual terms combine into one constant. For example, \((- \cos x+C_1)+( \sin x+C_2)\) can be written as \(-\cos x+\sin x+C\), where \(C=C_1+C_2\). There is no need to keep a separate arbitrary constant for each term.