Integrating with Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Integrating with reciprocal trigonometric functions

cosec (cosecant, csc), sec (secant) and cot (cotangent) are the reciprocal functions of sine, cosine and tangent respectively.

What are the antiderivatives involving reciprocal trigonometric functions?

• $\int {\sec }^{2}xdx=\tan x+c$

• $\int \sec x\tan xdx=\sec x+c$

• $\int \mathrm{cosec}x\cot xdx=-\mathrm{cosec}x+c$

• $\int {\mathrm{cosec}}^{2}xdx=-\cot x+c$

How do I integrate these if a linear function of x is involved?

• All integration rules could apply alongside the results above

• The use of reverse chain rule is particularly common

  • For linear functions the following results can be useful

    • $\int {\sec }^2\left(ax+b\right)\mathrm{d}x=\frac{1}{a}\tan \left(ax\mathit{+}b\right)+c$

    • $\int \sec \left(ax+b\right)\tan \left(ax+b\right)dx=\frac{1}{a}\sec \left(ax+b\right)+c$

    • $\int \mathrm{cosec}\left(ax+b\right)\cot \left(ax+b\right)dx=-\frac{1}{a}\mathrm{cosec}\left(ax+b\right)+c$

    • $\int {\mathrm{cosec}}^2\left(ax+b\right)dx=-\frac{1}{a}\cot \left(ax+b\right)+c$