Mr. Integral
Math notes/Calculus I/The Integral

THE WHOLE UNIT · ONE REFERENCE

The Integral
Cheat sheet.

The key rules, formulas and reminders from all 5 topics, gathered into reference cards.

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Key formulas, conditions and traps · Read down each column.

Antiderivatives and Initial Values

Core rule

An antiderivative satisfies F′=fF'=f. All antiderivatives on one interval are F+CF+C.

∫xpdx=xp+1/(p+1)+C (p≠−1),\int x^pdx=x^{p+1}/(p+1)+C\ (p\ne-1), ∫dx/x=ln⁡∣x∣+C.\int dx/x=\ln|x|+C.

Watch for

Each integration introduces a constant. Apply initial conditions afterward and verify by differentiation. Disconnected intervals can have independent constants.

Riemann Sums and Accumulation

Core rule

Δx=(b−a)/n,\Delta x=(b-a)/n, ∫abf(x)dx=lim⁡n→∞∑i=1nf(xi∗)Δx.\int_a^bf(x)dx=\lim_{n\to\infty}\sum_{i=1}^nf(x_i^*)\Delta x.

Watch for

Include interval widths. Left/right bracketing requires monotonicity. Below-axis contributions are negative; total area integrates ∣f∣|f|.

Definite Integrals and the Fundamental Theorem

Core rule

∫abf=F(b)−F(a),\int_a^bf=F(b)-F(a), ddx∫ag(x)f(t)dt=f(g(x))g′(x).\frac d{dx}\int_a^{g(x)}f(t)dt=f(g(x))g'(x).

For continuous ff, average value is 1b−a∫abf\frac1{b-a}\int_a^bf.

Watch for

Check continuity or justified integrability and the antiderivative hypotheses. Keep variable-bound chain factors; never skip an interior singularity.

Substitution and Transformed Bounds

Core rule

∫f(g(x))g′(x)dx=F(g(x))+Cwhen F′=f.\int f(g(x))g'(x)dx=F(g(x))+C\quad\text{when }F'=f.

Watch for

Rewrite all variable factors. For definite integrals, either transform the bounds or return to the original variable before evaluation. Track orientation and domains.

Net Change, Distance and a Calculus I Checkpoint

Core rule

q(b)=q(a)+∫abq′(t)dt.q(b)=q(a)+\int_a^bq'(t)dt.

Displacement integrates velocity; distance integrates its absolute value. Average rate is net change divided by elapsed time.

Watch for

Split at velocity sign changes, include the initial amount, preserve units and respect physical domain constraints.

01

Antiderivatives and Initial Values

2 reference blocks

Read lesson ↗

Core rule

An antiderivative satisfies F′=fF'=f. All antiderivatives on one interval are F+CF+C.

∫xpdx=xp+1/(p+1)+C (p≠−1),\int x^pdx=x^{p+1}/(p+1)+C\ (p\ne-1), ∫dx/x=ln⁡∣x∣+C.\int dx/x=\ln|x|+C.

Watch for

Each integration introduces a constant. Apply initial conditions afterward and verify by differentiation. Disconnected intervals can have independent constants.

02

Riemann Sums and Accumulation

2 reference blocks

Read lesson ↗

Core rule

Δx=(b−a)/n,\Delta x=(b-a)/n, ∫abf(x)dx=lim⁡n→∞∑i=1nf(xi∗)Δx.\int_a^bf(x)dx=\lim_{n\to\infty}\sum_{i=1}^nf(x_i^*)\Delta x.

Watch for

Include interval widths. Left/right bracketing requires monotonicity. Below-axis contributions are negative; total area integrates ∣f∣|f|.

03

Definite Integrals and the Fundamental Theorem

2 reference blocks

Read lesson ↗

Core rule

∫abf=F(b)−F(a),\int_a^bf=F(b)-F(a), ddx∫ag(x)f(t)dt=f(g(x))g′(x).\frac d{dx}\int_a^{g(x)}f(t)dt=f(g(x))g'(x).

For continuous ff, average value is 1b−a∫abf\frac1{b-a}\int_a^bf.

Watch for

Check continuity or justified integrability and the antiderivative hypotheses. Keep variable-bound chain factors; never skip an interior singularity.

04

Substitution and Transformed Bounds

2 reference blocks

Read lesson ↗

Core rule

∫f(g(x))g′(x)dx=F(g(x))+Cwhen F′=f.\int f(g(x))g'(x)dx=F(g(x))+C\quad\text{when }F'=f.

Watch for

Rewrite all variable factors. For definite integrals, either transform the bounds or return to the original variable before evaluation. Track orientation and domains.

05

Net Change, Distance and a Calculus I Checkpoint

2 reference blocks

Read lesson ↗

Core rule

q(b)=q(a)+∫abq′(t)dt.q(b)=q(a)+\int_a^bq'(t)dt.

Displacement integrates velocity; distance integrates its absolute value. Average rate is net change divided by elapsed time.

Watch for

Split at velocity sign changes, include the initial amount, preserve units and respect physical domain constraints.