Trigonometric Integrals

Trigonometric functions appear in the natural world frequently, so understanding how to integrate them is essential to solving real problems. There are a number of situations we can have involving trigonometric function, and corresponding rules to solve them.

Example: Evaluate \int cos^3(x)dx

If we tried using the substitution rule, we would have u = cos(x) and du = -sin(x)dx, which doesn’t really help. However, we can manipulate this function to be easier to work with. If we factor our a cos^2(x) term, we would have \int (cos^2(x)cos(x))dx

From here, we can use the trig identity, sin^2(x) + cos^2(x) = 1 to change the cos^2(x) into 1-sin^2(x). This gives us the following integral.

\int(1-sin^2(x))(cos(x))dx

From here, we can now apply the substitution rule to get a solution. If we let u = sin(x), then du = cos(x)dx, giving us the integral \int (1-u^2)du = u - \frac{1}{3}u^3 + C = sin(x) - \frac{1}{3}sin^3(x) + C

When we have functions in the form sin^m(x)cos^n(x), there is a set of rules we can follow to evaluate them correctly.

Strategy for evaluating \int sin^m(x)cos^n(x)

  1. If the power of cosine, n, is odd (n = 2k+1), save one factor of cosine and use cos^2(x) = 1-sin^2(x) to express the remaining factors in terms of sine: \int sin^mcos^{2k+1}dx = \int sin^m(x)(cos^2(x))^kcos(x) dx = \int sin^m(x)(1-sin^2(x))^kcos(x) dx, then substitute u = sin(x)
  2. If the power of sine is odd (m = 2k+1), save one factor of sine and use sin^2(x) = 1 - cos^2(x) to express the remaining factors in terms of cosine: \int sin^{2k+1}(x)cos^n dx = \int (sin^2(x))^kcos^n(x)sin(x)dx = \int (1-cos^2(x))^k cos^n(x) sin(x) dx, then substitute u = cos(x)
  3. If both the power of sine is odd and the power of cosine is odd, then either 1 or 2 can be used.
  4. If the powers of both sine and cosine are even, use the half angle identities sin^2(x) = \frac{1}{2}(1-cos(2x)) or cos^2(x) = \frac{1}{2}(1+cos(2x)

We can also evaluate functions that include tan(x) and sec(x)

Strategy for evaluating \int tan^m(x)sec^n(x) dx

  1. If the power of secant is even (n = 2k), save a factor of sec^2(x), and use sec^2(x) = 1+tan^2(x) to express the remaining factors in terms of tan(x): \int tan^m sec^{2k} dx = \int tan^m(sec^2(x))^{k-1}sec^2(x) dx = \int tan^m(x) (1+tan^2(x))^{k-1} sec^2(x) dx, the substitute u = tan(x)
  2. If the power of tangent is odd (m = 2k+1), save a factor of sec(x)tan(x) and use tan^2(x) = sec^2(x) - 1 to express the remaining factors in terms of sec(x): \int tan^{2k+1}(x)sec^n(x) dx
    = \int (tan^2(x))sec^{n-1}(x) sec(x)tan(x) dx
    = \int (sec^2(x) - 1)^k sec^{n-1}(x)tan(x) dx, then substitute u = sec(x).

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