Integral Calculator
Essential Calculus Resources

Common Integrals
Formulas & Examples

Your go-to cheat sheet for the most common integrals found in calculus. From basic powers to complex trigonometric functions.

Mastering Common Integrals

Memorizing or knowing how to derive these common integrals is crucial for success in Calculus I and II. Below is a comprehensive list covering trigonometric, exponential, logarithmic, and inverse trigonometric functions.

Basic & Exponential Integrals

Integral of x²

∫ x² dx = x³/3 + C

Rule: Power Rule. Increase the exponent by 1 and divide by the new exponent.

Integral of e^(2x)

∫ e^(2x) dx = ½e^(2x) + C

Rule: u-substitution. Let u = 2x, then du = 2dx.

Logarithmic Integrals

Integral of ln(x)

∫ ln(x) dx = x·ln(x) - x + C

Method: Integration by Parts.

  • Let u = ln(x), dv = dx
  • Then du = (1/x)dx, v = x
  • ∫ u dv = uv - ∫ v du = x·ln(x) - ∫ x(1/x) dx = x·ln(x) - x

Trigonometric Integrals

Integral of sin(x)

∫ sin(x) dx = -cos(x) + C

Integral of cos(x)

∫ cos(x) dx = sin(x) + C

Integral of tan(x)

∫ tan(x) dx = -ln|cos(x)| + C

Hint: Rewrite tan(x) as sin(x)/cos(x) and use u-substitution (u = cos(x)).

Integral of sec(x)

∫ sec(x) dx = ln|sec(x) + tan(x)| + C

Hint: Multiply numerator and denominator by (sec(x) + tan(x)).

Integral of sec²(x)

∫ sec²(x) dx = tan(x) + C

Note: This is a standard derivative rule in reverse (d/dx tan(x) = sec²(x)).

Advanced Trigonometric Integrals

Integral of sin²(x)

∫ sin²(x) dx = x/2 - sin(2x)/4 + C

Method: Use the half-angle identity: sin²(x) = (1 - cos(2x))/2.

Integral of cos²(x)

∫ cos²(x) dx = x/2 + sin(2x)/4 + C

Method: Use the half-angle identity: cos²(x) = (1 + cos(2x))/2.

Inverse Trigonometric Integrals

Integral of arcsin(x)

∫ arcsin(x) dx = x·arcsin(x) + √(1-x²) + C

Method: Integration by Parts (u = arcsin(x), dv = dx).

Integral of arctan(x)

∫ arctan(x) dx = x·arctan(x) - ½ln(1+x²) + C

Method: Integration by Parts (u = arctan(x), dv = dx).