Integrating with Trigonometric Identities

The Pythagorean identities serve as the most fundamental tools for swapping between squared trigonometric terms, specifically $\sin^2 x + \cos^2 x = 1$ and its related forms $1 + \tan^2 x = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$. These are essential because while we can integrate $\sec^2 x$ directly to get $\tan x$, we cannot directly integrate $\tan^2 x$ without this conversion.

Double angle identities for $\cos 2x$ are used to reduce the power of sine and cosine terms from squared to linear. By rearranging $\cos 2x = 1 - 2\sin^2 x$ and $\cos 2x = 2\cos^2 x - 1$, we can express $\sin^2 x$ and $\cos^2 x$ as functions of $\cos 2x$, which are straightforward to integrate via the reverse chain rule.

To integrate $\tan^2 kx$, replace it with $(\sec^2 kx - 1)$. This converts the expression into a standard derivative term ($\sec^2 kx$) and a constant ($-1$), resulting in $(\frac{1}{k}\tan kx - x + c)$.

When faced with $\sin^2 kx$, apply the identity $\sin^2 kx = \frac{1}{2}(1 - \cos 2kx)$. Integration then yields $\frac{1}{2}(x - \frac{1}{2k}\sin 2kx) + c$. This methodology relies on the fact that integrating $\cos 2kx$ is a simple reverse chain rule application.

For $\cos^2 kx$, the substitution is $\cos^2 kx = \frac{1}{2}(1 + \cos 2kx)$. Note that the only difference between the sine and cosine squared substitutions is the sign before the cosine term in the identity.

Check the Booklet First: Always verify standard results and identities in the formula booklet to avoid simple sign errors or coefficient mistakes. Standard integrals for $\sec^2 x$ and $\tan x$ are frequently provided and should be used as a baseline.

Look for a Two-Part Structure: Examiners often structure questions where part (a) asks for an identity proof and part (b) requires its integration. If you fail to prove the identity, use the 'show that' result to attempt the integration anyway to secure method marks.

Verify by Differentiation: If time permits, differentiate your final answer. If you integrated $\sin^2 x$ and your result differentiates back to $\sin^2 x$ (using the same identities in reverse), you can be certain of your accuracy.

Incorrect Integration of Powers: A frequent error is treating $\int \sin^2 x \, dx$ like a polynomial power, resulting in $\frac{\sin^3 x}{3}$. This is incorrect because the chain rule would require a $\cos x$ term to be present in the original integrand for this to work.

Coefficient Confusion: When integrating $\cos 2kx$, students often forget to divide by the coefficient $2k$ (the derivative of the angle). Always apply the compensation factor $\frac{1}{a}$ when integrating $f(ax + b)$.

Sign Errors in $\cos 2x$: Mixing up the signs between the $\sin^2 x$ and $\cos^2 x$ identities is a common source of lost marks. Remember that $\cos 2x$ is subtracted to represent the square of the sine function.