Integrals of tan, cot, sec, cosec
\[\int\tan x=\log|\sec x|,\ \int\cot x=\log|\sin x|,\ \int\sec x=\log|\sec x+\tan x|,\ \int\csc x=\log|\csc x-\cot x|\]
Integrals
Integration by Substitution
Understand
If the integrand contains a function \(g(x)\) together with its derivative \(g'(x)\), substitute \(u=g(x)\), \(du=g'(x)\,dx\):
\[\int f(g(x))\,g'(x)\,dx=\int f(u)\,du.\]
Example: \(\int2x\cos(x^2)\,dx\), put \(u=x^2\): \(\int\cos u\,du=\sin(x^2)+C\).
Important consequence: \(\int\frac{g'(x)}{g(x)}\,dx=\log|g(x)|+C\). This gives \(\int\tan x\,dx=\log|\sec x|+C\), \(\int\cot x\,dx=\log|\sin x|+C\), \(\int\sec x\,dx=\log|\sec x+\tan x|+C\) and \(\int\csc x\,dx=\log|\csc x-\cot x|+C\).
Also useful: \(\int f(ax+b)\,dx=\frac1aF(ax+b)+C\).
Key Concepts
- Look for a function and its derivative.
- (numerator) = derivative of (denominator) ⇒ log.
- Always return to the original variable.
Formula Bank
Key Points
Numerator = derivative of denominator
\(\int\frac{f\,'(x)}{f(x)}dx=\log|f(x)|+C\), e.g. \(\int\frac{2x+3}{x^2+3x+1}dx=\log|x^2+3x+1|+C\).
Common Mistakes
Forgetting to change limits
After substituting u = g(x) in a definite integral, either change the limits to u-values or substitute back before using the original limits.
Practice & Topic Test
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Choose u so that du appears in the integrand; change everything (including dx) into u; integrate; substitute back.
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