Indefinite integrals, antiderivatives, and linearity

Key idea

An indefinite integral represents a family of antiderivatives: \[ \int f(x)\,dx = F(x) + C \quad \text{where } F'(x)=f(x). \] The constant of integration \(+C\) is required because differentiating any constant gives \(0\).

Linearity rules you’ll use constantly

• Constant multiple: \(\int c\,f(x)\,dx = c\int f(x)\,dx\)
• Sum/Difference: \(\int (f(x)\pm g(x))\,dx = \int f(x)\,dx \pm \int g(x)\,dx\)

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Summary

• \(\int f(x)\,dx = F(x)+C\) where \(F'(x)=f(x)\).
• Use linearity to break integrals into simpler pieces.

The power rule for integration and the \(\int \frac{1}{x}\,dx\) exception

Key idea

The most-used rule for antiderivatives is the power rule: \[ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C \quad (n\neq -1). \] The exponent increases by \(1\), then you divide by the new exponent.

But when \(n=-1\), the formula would divide by \(0\). That special case is: \[ \int \frac{1}{x}\,dx=\ln|x|+C. \]

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Summary

• Power rule: \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\) for n≠ -1.
• Special case: \(\int \dfrac{1}{x}\,dx=\ln|x|+C\).

Common trigonometric and inverse trigonometric integrals

Core trig integrals to know cold

• \(\displaystyle \int \cos x\,dx=\sin x + C\)
• \(\displaystyle \int \sin x\,dx=-\cos x + C\)
• \(\displaystyle \int \sec^2 x\,dx=\tan x + C\)
• \(\displaystyle \int \csc^2 x\,dx=-\cot x + C\)
• \(\displaystyle \int \sec x\tan x\,dx=\sec x + C\)
• \(\displaystyle \int \csc x\cot x\,dx=-\csc x + C\)

Inverse trig pattern

A must-know inverse trig antiderivative is: \[ \int \frac{1}{1+x^2}\,dx=\arctan(x)+C. \] This is the antiderivative because \(\dfrac{d}{dx}\arctan(x)=\dfrac{1}{1+x^2}\).

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Summary

• Know the standard trig antiderivatives and the sign patterns.
• \(\int \dfrac{1}{1+x^2}\,dx=\arctan(x)+C\) is a key inverse trig result.

u-substitution: the reverse chain rule for integrals

Key idea

If you can identify a composition \(f(g(x))\) together with \(g'(x)\), then the integral often simplifies by substituting \(u=g(x)\). In practice, you’re matching a derivative pattern: \[ \int f(g(x))\,g'(x)\,dx = \int f(u)\,du. \] You do not need full algebraic substitution for every problem — many early integrals are “pattern matches.”

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Summary

• Look for an inside function \(u=g(x)\) and (a constant multiple of) its derivative \(g'(x)\,dx\).
• u-substitution is the reverse chain rule for integrals.

Definite integrals and the Fundamental Theorem of Calculus

Key idea

A definite integral measures net accumulation from \(a\) to \(b\): \[ \int_a^b f(x)\,dx. \] If \(F'(x)=f(x)\), then the Fundamental Theorem of Calculus says: \[ \int_a^b f(x)\,dx = F(b)-F(a). \] Notice: there is no \(+C\) in a definite integral because the constants cancel when you subtract.

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Summary

• Definite integrals use evaluation: \(\int_a^b f(x)\,dx=F(b)-F(a)\).
• No \(+C\) in definite integrals (constants cancel).

Mixed integration patterns: powers, rationals, and quick recognition

Key idea

Most early integration problems are about picking the right pattern fast:

• Power rule: \(x^n\) with n≠ -1
• Log form: \(\dfrac{1}{x}\) or \(\dfrac{g'(x)}{g(x)}\)
• Trig identities: \(\sec^2 x\), \(\csc^2 x\), \(\sec x\tan x\), \(\csc x\cot x\)
• u-substitution: something like \((\text{inside})'\cdot(\text{inside})^n\)

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Summary

• Practice is about fast pattern recognition: power rule vs. log form vs. trig vs. u-substitution.
• Always include \(+C\) for indefinite integrals.

Why integrals and antiderivatives matter

Where integrals show up

• Area and net area: \(\int_a^b f(x)\,dx\) measures signed area / net change.
• Accumulation: total change from a rate function (physics, economics, biology).
• Probability: continuous distributions use integrals for total probability.
• Inverse of differentiation: antiderivatives “undo” derivatives.

Worked example: from antiderivative to definite integral

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Final recap

• Indefinite integrals: \(\int f(x)\,dx=F(x)+C\) where \(F'(x)=f(x)\).
• Power rule: \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\) for n≠ -1.
• Log case: \(\int \dfrac{1}{x}\,dx=\ln|x|+C\).
• Trig basics: \(\int \sec^2x\,dx=\tan x+C\), \(\int \csc^2x\,dx=-\cot x+C\), \(\int \csc x\cot x\,dx=-\csc x+C\).
• u-substitution: reverse chain rule, especially \(\int \dfrac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C\).
• Definite integrals: \(\int_a^b f(x)\,dx=F(b)-F(a)\) (no \(+C\)).