Use trigonometric identities

Trigonometric standard forms and identities

Check the inner derivative before choosing a formula. If no direct match appears, simplify with an identity: lower even sine/cosine powers, or replace tangent/cotangent squares using the corresponding secant/cosecant identity. Recheck signs and chain-rule factors after integrating.

Reverse $\frac{d}{dx}\sin[f(x)]=f^{\prime}(x)\cos[f(x)]$ to get the first composite rule below. The same reasoning gives all six source pairs (each with an arbitrary $C$):

Basic integral: $\int\cos x\,dx=\sin x+C$

Composite integral: $\int f^{\prime}(x)\cos[f(x)]\,dx=\sin[f(x)]+C$

Basic integral: $\int\sin x\,dx=-\cos x+C$

Composite integral: $\int f^{\prime}(x)\sin[f(x)]\,dx=-\cos[f(x)]+C$

Basic integral: $\int\sec^2x\,dx=\tan x+C$

Composite integral: $\int f^{\prime}(x)\sec^2[f(x)]\,dx=\tan[f(x)]+C$

Basic integral: $\int\csc^2x\,dx=-\cot x+C$

Composite integral: $\int f^{\prime}(x)\csc^2[f(x)]\,dx=-\cot[f(x)]+C$

Basic integral: $\int\sec x\tan x\,dx=\sec x+C$

Composite integral: $\int f^{\prime}(x)\sec[f(x)]\tan[f(x)]\,dx=\sec[f(x)]+C$

Basic integral: $\int\csc x\cot x\,dx=-\csc x+C$

Composite integral: $\int f^{\prime}(x)\csc[f(x)]\cot[f(x)]\,dx=-\csc[f(x)]+C$

The source writes cosec for csc. The Teacher labels the basic forms O-level and the composite forms H2-level, emphasising the inner derivative and the minus signs in the sine/cosecant cases; the six composite rules are annotated as not in MF27.

Four logarithmic formulas are marked available in MF27: $\int\tan x\,dx=\ln(\sec x)+C$ and $\int\sec x\,dx=\ln(\sec x+\tan x)+C$ on $|x|<\pi/2$; $\int\cot x\,dx=\ln(\sin x)+C$ and $\int\csc x\,dx=-\ln(\csc x+\cot x)+C$ on $0<x<\pi$. The source remark extends these using absolute values inside the logarithms; work on connected intervals where the original trigonometric functions are defined.

For even powers of sine/cosine, lower the power with $\sin2A=2\sin A\cos A$ and $\cos2A=2\cos^2A-1=1-2\sin^2A$, marked MF27. Rearranging gives $\cos^2A=(1+\cos2A)/2$ and $\sin^2A=(1-\cos2A)/2$. Examples 12–13 use these, applying double angle twice for the fourth power. The Teacher crosses out the false power-rule answer $\cos^3x/3+C$ to $\int\cos^2x\,dx$: its derivative is $-\cos^2x\sin x$, not $\cos^2x$.

Other printed identities are $\cos^2A+\sin^2A=1$, $1+\tan^2A=\sec^2A$ and $1+\cot^2A=\csc^2A$. The Teacher adds a worked demonstration: $\int\tan^2x\,dx=\int(\sin^2x/\cos^2x)\,dx=\int(\sec^2x-1)\,dx=\tan x-x+C$, with $\frac{d}{dx}\tan x=\sec^2x$. Keep $\cos x\ne0$ on its interval. This is a Teacher annotation, not a new printed Student exercise. Example 15 retains both supplied product-to-sum identities.