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Anti-Derivative Rules
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Basic anti-derivative rules for common functions: ∫xndx=n+1xn+1+C for n=−1, ∫sinxdx=−cosx+C, ∫cosxdx=sinx+C, ∫exdx=ex+C, ∫x1dx=ln∣x∣+C, and ∫sec2xdx=tanx+C.
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Anti-Derivative of xⁿ, n∈ℚ
AHL AI 5.11
∫xndx=n+1xn+1+C,n=−1📖
Anti-Derivative of e^x
AHL AI 5.11
∫exdx=ex+C📖
Anti-Derivative of sin and cos
AHL AI 5.11
The integrals of sin and cos are
∫sinxdx ∫cosxdx=−cosx+C📖 =sinx+C📖
Anti-Derivative of 1/x
AHL AI 5.11
∫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 AI 5.11
∫sec2xdx=tanx+C
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