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Anti-Derivative Rules
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Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\), \(\int \sin x\,dx=-\cos x+C\), \(\int \cos x\,dx=\sin x+C\), \(\int e^x\,dx=e^x+C\), \(\int \frac{1}{x}\,dx=\ln|x|+C\), and the usual trig and inverse trig results.
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Anti-Derivative of xⁿ, n∈ℚ
SL 5.10
∫xndx=n+1xn+1+C,n=−1📖
Anti-Derivative of e^x
SL 5.10
∫exdx=ex+C📖
Anti-Derivative of sin and cos
SL 5.10
The integrals of sin and cos are
∫sinxdx ∫cosxdx=−cosx+C📖 =sinx+C📖
Anti-derivatives of sums and scalar multiples
SL 5.10
The anti-derivative of a sum is the sum of the anti-derivatives:
∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx
The anti-derivative of a scalar multiple (a constant that does not depend on the integrating variable) can pass through the integral:
∫kf(x)dx=k∫f(x)dx
Anti-Derivative of 1/x
SL 5.6
∫x1dx=ln∣x∣+C📖
Why the absolute value?
x1 is defined for x<0, but lnx is not. Specifically:
dxd(ln(−x))=−x−1=x1,
So x1 is the derivative of lnx and of ln(−x).
However, we can simplify further. Recall the definition of the absolute value:
∣x∣={−xx<0xx>0.
Hence, we have
dxd(ln∣x∣)=x1
⟹∫x1dx=ln(∣x∣)+C.
Anti-Derivative leading to tan
AHL 5.15
∫sec2xdx=tanx+C
Anti-Derivative leading to sec, cosec, and cot
AHL 5.15
∫secxtanxdx−∫cosecxcotx−∫cosec2x=secx+Cdx=cosecx+Cdx=cotx+C
Anti-Derivatives leading to arccos, arcsin, or arctan
AHL 5.15
∫√a2−x2−1dx=arccos(ax)+C📖
∫√a2−x21dx=arcsin(ax)+C📖
∫a2+x21dx=a1arctan(ax)+C📖
Anti-Derivative of aˣ
AHL 5.15
∫axdx=lna1ax+C📖
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