Indefinite Integrals: A Calculus Guide

Indefinite Integrals with Constants

• The anti-derivative of a constant involves adding an x variable to it,
• Remember to add the "+ c" constant of integration to represent any constant value.
• If integrating π with respect to y, the result is πy + c.
• Integrating 'e' with respect to z yields ez + c.

Integrating a Variable Raised to a Constant

• The rule for integrating x^n is (x^(n+1))/(n+1) + c.
• For example, the anti-derivative of x^2 is x^3/3 + c.
• When integrating 8x^3, the result is 2x^4 + c.
• The anti-derivative of 5x^6 is (5/7)x^7 + c.
• Integrating 7x (or 7x^1) results in (7/2)x^2 + c.
• The anti-derivative of 3x is (3/2)x^2 + c.

Integrating Polynomials

• Integrate each term separately.
• The anti-derivative of x^2 - 5x + 6 is x^3/3 - (5/2)x^2 + 6x + c.
• The anti-derivative of 4x^3 + 8x^2 - 9 is x^4 + (8/3)x^3 - 9x + c.

Integrating Square Root Functions

• Rewrite the square root of x as x^(1/2).
• The anti-derivative of x^(1/2) is (2/3)x^(3/2) + c.
• Rewrite the cube root of x^4 as x^(4/3).
• The anti-derivative of x^(4/3) is (3/7)x^(7/3) + c, which can also be written as (3/7)(cube root of x^7) + c.

Integrating Squared Expressions

• Expand the expression first (e.g., (3x - 1)^2 = 9x^2 - 6x + 1) before integrating.
• The anti-derivative of 9x^2 - 6x + 1 is 3x^3 - 3x^2 + x + c.

Integrating the Product of Polynomials

• Expand the expression first.
• The expression (2x + 1)(x - 2) becomes 2x^2 - 3x - 2.
• The anti-derivative of 2x^2 - 3x - 2 is (2/3)x^3 - (3/2)x^2 - 2x + c.

Integrating Fractions

• If there's a single term in the denominator, separate the fraction into smaller fractions.
• For (x^4 + 6x^3)/x, separate it into x^3 + 6x^2 before integrating.
• The anti-derivative of x^3 + 6x^2 is (1/4)x^4 + 2x^3 + c.

Integrating Expressions with Variables or exponents in the Denominator

• Rewrite the fraction with a negative exponent, 1/x^2 becomes x^(-2).
• The anti-derivative of x^(-2) is -1/x + c.
• The anti-derivative of 1/x^3 is -1/(2x^2) + c.
• The anti-derivative of 5/x^4 is -5/(3x^3) + c.

Integrating 1/x

• The anti-derivative of 1/x is ln|x| + c.
• The anti-derivative of 1/(x - 3) is ln|x - 3| + c.
• The anti-derivative of 1/(x + 4) is ln|x + 4| + c.
• The anti-derivative of 5/(x - 2) is 5ln|x - 2| + c.

Integrating Exponential Functions

• The anti-derivative of e^(4x) is e^(4x)/4 + c.
• The same logic works for values other than 4
• The anti-derivative of e^x is e^x + c.
• The anti-derivative of 8e^(2x) is 4e^(2x) + c.
• The anti-derivative of 12e^(3x) is 4e^(3x) + c.

Integrating Trigonometric Functions

• The anti-derivative of cos(x) is sin(x) + c.
• The anti-derivative of sin(x) is -cos(x) + c.
• The anti-derivative of cos(3x) is sin(3x)/3 + c.
• The anti-derivative of cos(7x) is sin(7x)/7 + c.
• The anti-derivative of 14sin(2x) is -7cos(2x) + c.
• The anti-derivative of 6sin(3x) is -2cos(3x) + c.

Integrating Secant Functions

• The anti-derivative of secant^2(x) is tan(x) + c.
• The anti-derivative of 8secant^2(4x) is 2tan(4x) + c.
• The anti-derivative of secant(x)tan(x) is secant(x) + c.
• The anti-derivative of 12secant(3x)tan(3x) is 4secant(3x) + c.

U-Substitution

• Used when there is a more complex term in the integrand
• Replace all x variables with u variables
• The goal is to simplify the integral by substituting a part of the function with 'u'.
• Choose 'u' such that its derivative simplifies the integral.
• If integrating x^2*sin(x^3), let u = x^3. Then du = 3x^2 dx, and dx = du/(3x^2).
• Substitute and solve the new integral. Remember to change back to the original variable, use the initial substitution definition
• After substituting, the integral becomes (1/3) * integral of sin(u) du, which is -(1/3)cos(u) + c.
• Replace 'u' to get the final answer: -(1/3)cos(x^3) + c.
• Steps:
  • Choose u and find du/dx.
  • Find dx in terms of du.
  • Substitute u and dx in the original integral.
  • Evaluate the new integral in terms of u.
  • Replace u with the original function of x.

U-Substitution, Example 2

• To integrate 5x(x^2 + 3)^4, let u = x^2 + 3. Then du = 2x dx, so dx = du / (2x).
• Substitute u and dx into the integral: integral of 5x * u^4 * (du / (2x)).
• Simplify and integrate: (5/2) * integral of u^4 du = (5/2) * (u^5 / 5) + c = (1/2) * u^5 + c.
• Substitute back for 'u': (1/2) * (x^2 + 3)^5 + c.

U-Substitution, Example 3

• To integrate tangent(x), rewrite it as sin(x) / cos(x).
• Let u = cos(x), so du = -sin(x) dx, and dx = du / (-sin(x)).
• Rewrite the integral in terms of u, leading to - integral of 1/u du, which is -ln|u| + c.
• Substituting back yields -ln|cos(x)| + c, which can be rewritten as ln|secant(x)| + c given log rules.

Integration by Parts

• Used when u-substitution doesn't work
• Use the formula: integral of u dv = uv - integral of v du.
• Choose 'u' and 'dv' strategically.
• Works well if a term in the equation will disappear when you find du
• For integrating x * cos(x) dx, let u = x and dv = cos(x) dx. Then du = dx and v = sin(x).
• Use the formula to rewrite the integral as x*sin(x) - integral of sin(x) dx.
• Completing the integration yields x*sin(x) + cos(x) + c.
• The goal is to transform a difficult integral into a simpler one.
• Steps:
  • Choose u and dv.
  • Find du (derivative of u) and v (integral of dv).
  • Apply the formula: ∫ u dv = uv - ∫ v du.
  • Evaluate the new integral.

Integration by Parts, Example 2

• With x * e^(4x) dx, let u = x and dv = e^(4x) dx. Thus, du = dx and v = (1/4)e^(4x).
• Apply the integration by parts formula to get (1/4)xe^(4x) - integral of (1/4)e^(4x) dx.
• Solve the remaining integral to get (1/4)xe^(4x) - (1/16)e^(4x) + c.

Trigonometric Substitution

• Use when u-substitution OR integration by parts do not work
• The goal is to eliminate or simplify the square root.
• Used to integrate expressions involving square roots of a^2 - x^2, a^2 + x^2, or x^2 - a^2.
• Requires knowledge of trig identities
• For integrating 4 / (1 + x^2) dx, use the identity 1 + tan^2(θ) = secant^2(θ).
• Let x = tangent(θ), so dx = secant^2(θ) dθ. Then 1 + x^2 becomes 1 + tan^2(θ), which is secant^2(θ).
• Substitute and simplify to get 4 * integral of dθ, which is 4θ + c.
• Since x = tangent(θ), θ = arctangent(x) aka "atan(x)", so the final answer is 4 arctangent(x) + c.

Trigonometric Substitution, Example 2

• Inegrating 3 / sqrt(1 - x^2) dx, use the identity 1 - sin^2(θ) = cos^2(θ).
• Let x = sin(θ), so dx = cos(θ) dθ. Then 1 - x^2 becomes 1 - sin^2(θ), which is cos^2(θ).
• The integral simplifies to 3 * integral of dθ, which is 3θ + c.
• Since x = sin(θ), θ = arcsin(x), aka "asin(x)" so the final answer is 3arcsin(x) + c.