Antiderivatives Guide

The integral of a constant is the constant times x.

Example: ∫5 dx = 5x + C

For any real number n ≠ -1.

Example: ∫x³ dx = x⁴/4 + C

The integral of a sum is the sum of the integrals.

Example: ∫(x² + 3x) dx = ∫x² dx + ∫3x dx

Constants can be factored out of integrals.

Example: ∫4x³ dx = 4∫x³ dx = 4(x⁴/4) + C = x⁴ + C

For positive integer exponents:

∫x³ dx = x⁴/4 + C

∫x⁵ dx = x⁶/6 + C

The exponent increases by 1, then divide by the new exponent.

For negative exponents (except -1):

∫x⁻² dx = x⁻¹/(-1) + C = -1/x + C

∫x⁻⁴ dx = x⁻³/(-3) + C = -1/(3x³) + C

Same rule applies: increase exponent by 1, then divide.

For fractional exponents:

∫x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3)x^(3/2) + C

∫x^(-1/2) dx = x^(1/2)/(1/2) + C = 2√x + C

Works the same way with fractions.

When n = -1, the power rule doesn't apply:

∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C

This is a special case that requires the natural logarithm.

Choose u to be the inner function of a composition.

Example: For ∫2x·e^(x²) dx, let u = x²

The derivative du/dx = 2x appears as a factor.

Differentiate u with respect to x to find du.

Example: u = x², so du = 2x dx

This allows us to substitute du for g'(x)dx.

Replace all x terms with u terms in the integral.

Example: ∫2x·e^(x²) dx = ∫e^u du

The integral becomes simpler in terms of u.

Integrate with respect to u, then substitute back to x.

Example: ∫e^u du = e^u + C = e^(x²) + C

Don't forget the constant of integration.

Derived from the product rule:

d(uv) = u dv + v du

Integrating both sides gives:

uv = ∫u dv + ∫v du

Rearranging gives the integration by parts formula.

Use the LIATE rule to choose u:

L - Logarithmic functions

I - Inverse trigonometric functions

A - Algebraic functions

T - Trigonometric functions

E - Exponential functions

Sometimes integration by parts needs to be applied multiple times.

This happens when the new integral ∫v du is similar to the original.

Examples include ∫eˣ sin(x) dx and ∫x² eˣ dx.

For repeated integration by parts, use the tabular method.

Create a table with derivatives of u and integrals of dv.

Multiply diagonally and alternate signs.

∫sin(x) dx = -cos(x) + C

∫cos(x) dx = sin(x) + C

∫sec²(x) dx = tan(x) + C

∫csc²(x) dx = -cot(x) + C

∫sec(x)tan(x) dx = sec(x) + C

For odd powers: save one factor, convert the rest

For even powers: use half-angle identities

Examples:

∫sin³(x) dx = ∫sin²(x)sin(x) dx

∫sin²(x) dx = ∫(1-cos(2x))/2 dx

For even powers of secant: save sec²(x) dx

For odd powers of tangent: save sec(x)tan(x) dx

Use identity: 1 + tan²(x) = sec²(x)

Examples:

∫tan³(x)sec(x) dx

For integrals with √(a²-x²): use x = a sin(θ)

For integrals with √(a²+x²): use x = a tan(θ)

For integrals with √(x²-a²): use x = a sec(θ)

Then use trigonometric identities to simplify.

The definite integral ∫[a,b] f(x) dx gives the area between the curve y=f(x) and the x-axis from x=a to x=b.

Example: Finding the area under a velocity-time graph gives displacement.

Used in physics, economics, and engineering.

Work: ∫F(x) dx gives work done by a variable force.

Displacement: ∫v(t) dt gives displacement from velocity.

Electric charge: ∫I(t) dt gives total charge from current.

Essential for solving physics problems.

Total cost: ∫MC(x) dx gives total cost from marginal cost.

Total revenue: ∫MR(x) dx gives total revenue from marginal revenue.

Consumer surplus: Area between demand curve and price.

Used in economic modeling and analysis.

Volume: ∫A(x) dx gives volume from cross-sectional area.

Population growth: ∫P'(t) dt gives population from growth rate.

Heat transfer: ∫q(t) dt gives total heat transferred.

Applied across all scientific disciplines.