Indefinite Integrals Summary

  • Indefinite Integrals Overview

    • The integral of a constant:
    • ∫4dx=4x+c\int 4 dx = 4x + c.
    • Always add a constant cc when integrating.

  • Basic Integrals

    • ∫πdy=πy+c\int \pi dy = \pi y + c
    • ∫edz=ez+c\int e dz = ez + c

  • Power Rule for Indefinite Integrals

    • ∫xndx=xn+1n+1+c\int x^n dx = \frac{x^{n+1}}{n+1} + c, where n≠−1n ≠ -1.
    • Example:
    • ∫x2dx=x33+c\int x^2 dx = \frac{x^3}{3} + c
    • ∫8x3dx=2x4+c\int 8x^3 dx = 2x^4 + c

  • Integrating Polynomials

    • Separate and integrate:
    • e.g. ∫(x2−5x+6)dx=x33−5x22+6x+c\int (x^2 - 5x + 6) dx = \frac{x^3}{3} - \frac{5x^2}{2} + 6x + c

  • Integrating Roots

    • Rewrite x\sqrt{x} as x1/2x^{1/2}.
    • Example:
    • ∫xdx=∫x1/2dx=23x3/2+c\int \sqrt{x} dx = \int x^{1/2} dx = \frac{2}{3}x^{3/2} + c

  • Integration Techniques

    • For fractions, separate terms:
    • ∫x4+6x3xdx=∫(x3+6x2)dx\int \frac{x^4 + 6x^3}{x} dx = \int (x^3 + 6x^2) dx

  • Trigonometric Integrals

    • ∫cos⁡(x)dx=sin⁡(x)+c\int \cos(x) dx = \sin(x) + c
    • ∫sin⁡(x)dx=−cos⁡(x)+c\int \sin(x) dx = -\cos(x) + c

  • Logarithmic Integrals

    • ∫1xdx=ln⁡∣x∣+c\int \frac{1}{x} dx = \ln|x| + c
    • Example: ∫5x−2dx=5ln⁡∣x−2∣+c\int \frac{5}{x - 2}dx = 5 \ln|x - 2| + c

  • Exponential Integrals

    • ∫ekxdx=1kekx+c\int e^{kx} dx = \frac{1}{k}e^{kx} + c
    • Example: ∫e4xdx=14e4x+c\int e^{4x} dx = \frac{1}{4}e^{4x} + c

  • Integration by Parts

    • Using the formula:
    • ∫udv=uv−∫vdu\int u dv = uv - \int v du
    • Example: For ∫xe4xdx\int x e^{4x} dx, choose:
      • u=xu = x, dv=e4xdxdv = e^{4x} dx

  • U-Substitution

    • For integrals like ∫x2sin⁡(x3)dx\int x^2 \sin(x^3)dx, let u=x3u = x^3, then du=3x2dxdu = 3x^2dx.

  • Trigonometric Substitution

    • For ∫41+x2dx\int \frac{4}{1+x^2}dx, use tangent: let x=tan⁡(θ)x = \tan(\theta). This simplifies integrations involving square roots.