Anti-Derivative Rules
Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as ∫xndx=n+1xn+1+C, ∫sinxdx=−cosx+C, ∫cosxdx=sinx+C, ∫exdx=ex+C, ∫x1dx=ln∣x∣+C, and the usual trig and inverse trig results.
Anti-Derivative of xⁿ, n∈ℚ
Anti-Derivative of e^x
Anti-Derivative of sin and cos
The integrals of sin and cos are
Anti-derivatives of sums and scalar multiples
The anti-derivative of a sum is the sum of the anti-derivatives:
The anti-derivative of a scalar multiple (a constant that does not depend on the integrating variable) can pass through the integral:
Anti-Derivative of 1/x
Why the absolute value?
x1 is defined for x<0, but lnx is not. Specifically:
So x1 is the derivative of lnx and of ln(−x).
However, we can simplify further. Recall the definition of the absolute value:
Hence, we have
Anti-Derivative leading to tan
Anti-Derivative leading to sec, cosec, and cot
Anti-Derivatives leading to arccos, arcsin, or arctan
Anti-Derivative of aˣ
Nice work completing Anti-Derivative Rules, here's a quick recap of what we covered:
Skills covered
Exercises checked off