Integration Formulas

Integration Formulas

The Power Rule

• Basic Integration Formulas:

  • ∫ 𝒂 𝒅𝒙 = 𝑎𝑥 + 𝑐

  • ∫ 𝒙ⁿ 𝒅𝒙 = 𝑥ⁿ⁺¹ / (𝑛 + 1) + 𝐶 (where n ≠ -1)

  • ∫(𝒂𝒙 + 𝒃)ⁿ 𝒅𝒙 = 1/𝑎 ∗ (𝑎𝑥 + 𝑏)ⁿ⁺¹ / (𝑛 + 1) + 𝐶

Rational Functions

• Integration of Rational Functions:

  • ∫ 1/𝑥 𝒅𝒙 = ln |𝑥| + 𝐶

  • ∫ 1/(𝑎𝑥 + 𝑏) 𝒅𝒙 = 1/𝑎 ln |𝑎𝑥 + 𝑏| + 𝐶

Exponential Functions

• Integration of Exponential Functions:

  • ∫ eˡˣ 𝒅𝒙 = eˡˣ + 𝐶

  • ∫ e^(𝑎𝑥 + 𝑏) 𝒅𝒙 = 1/𝑎 e^(𝑎𝑥 + 𝑏) + 𝐶

  • ∫ 𝑎^𝑏𝑥 + 𝑑 𝒅𝒙 = 𝑎𝑏𝑥 + 𝑑/𝑏 ln 𝑎 + 𝐶

Trigonometric Functions

• Integration of Trigonometric Functions:

  • ∫ sin 𝑥 𝒅𝒙 = -cos 𝑥 + 𝐶

  • ∫ sin(𝑎𝑥 + 𝑏) 𝒅𝒙 = -1/𝑎 cos(𝑎𝑥 + 𝑏) + 𝐶

  • ∫ cos 𝑥 𝒅𝒙 = sin 𝑥 + 𝐶

  • ∫ cos(𝑎𝑥 + 𝑏) 𝒅𝒙 = 1/𝑎 sin(𝑎𝑥 + 𝑏) + 𝐶

  • ∫ csc 𝑥 cot 𝑥 𝒅𝒙 = -csc 𝑥 + 𝐶

  • ∫ sec 𝑥 tan 𝑥 𝒅𝒙 = sec 𝑥 + 𝐶

  • ∫ tan 𝑥 𝒅𝒙 = ln |sec 𝑥| + 𝐶 = -ln |cos 𝑥| + 𝐶

  • ∫ cot 𝑥 𝒅𝒙 = ln |sin 𝑥| + 𝐶 = -ln |csc 𝑥| + 𝐶

  • ∫ sec 𝑥 𝒅𝒙 = ln |sec 𝑥 + tan 𝑥| + 𝐶

  • ∫ csc 𝑥 𝒅𝒙 = ln |csc 𝑥 - cot 𝑥| + 𝐶

Integration By Parts

• Formula for Integration by Parts:

  • ∫ 𝒖 𝒅𝒗 = 𝑢𝑣 - ∫ 𝑣 𝒅𝑢

  • Extended case: ∫ 𝒖𝒗 𝒅𝒘 = 𝑢𝑣𝑤 - ∫ 𝑣𝑤 𝒅𝑢 - ∫ 𝒖𝑤 𝒅𝑣

Integration By Parts - Related Formulas

• ∫ e^(𝑎𝑥) sin(𝑏𝑥) 𝒅𝒙 = (e^(𝑎𝑥)/(𝑎² + 𝑏²)) [𝑎 sin(𝑏𝑥) - 𝑏 cos(𝑏𝑥)] + 𝐶

• ∫ e^(𝑎𝑥) cos(𝑏𝑥) 𝒅𝒙 = (e^(𝑎𝑥)/(𝑎² + 𝑏²)) [𝑎 cos(𝑏𝑥) + 𝑏 sin(𝑏𝑥)] + 𝐶

• ∫ 𝑥ⁿ ln(𝑥) 𝒅𝒙 = (𝑥ⁿ⁺¹ / (𝑛 + 1)) ln(𝑥) - (𝑥ⁿ⁺¹ / (𝑛 + 1)²) + 𝐶 (𝑛 ≠ 1)

• ∫ 𝑥ⁿ e^(𝑎𝑥) 𝒅𝒙 = (𝑥ⁿ e^(𝑎𝑥)/𝑎) - (𝑛/𝑎) ∫ 𝑥ⁿ⁻¹ e^(𝑎𝑥) 𝒅𝑥

Trig Substitution

• **For Trig Substitution: **

  • ∫ √(𝑎² - 𝑢²) 𝒅𝑢 = (𝑢/2) √(𝑎² - 𝑢²) + (𝑎²/2) sin⁻¹(𝑢/𝑎) + 𝐶Substitution: 𝑢 = 𝑎 sin(𝜃) , 𝒅𝑢 = 𝑎 cos(𝜃) 𝒅𝜃

  • ∫ √(𝑎² + 𝑢²) 𝒅𝑢 = (𝑢/2) √(𝑢² + 𝑎²) + (𝑎²/2) ln |𝑢 + √(𝑢² + 𝑎²)| + 𝐶Substitution: 𝑢 = 𝑎 tan(𝜃) , 𝒅𝑢 = 𝑎 sec²(𝜃) 𝒅𝜃

  • ∫ √(𝑢² - 𝑎²) 𝒅𝑢 = (𝑢/2) √(𝑢² - 𝑎²) - (𝑎²/2) ln |𝑢 + √(𝑢² - 𝑎²)| + 𝐶Substitution: 𝑢 = 𝑎 sec(𝜃) , 𝒅𝑢 = 𝑎 sec(𝜃) tan(𝜃) 𝒅𝜃

Reduction Formulas

• ∫ sin²(𝑥) 𝒅𝑥 = (1/2) 𝑥 - (1/4) sin(2𝑥) + 𝐶 = (1/2)(𝑥 - sin 𝑥 cos 𝑥) + 𝐶

• ∫ cos²(𝑥) 𝒅𝑥 = (1/2) 𝑥 + (1/4) sin(2𝑥) + 𝐶 = (1/2)(𝑥 + sin 𝑥 cos 𝑥) + 𝐶

• ∫ tan²(𝑥) 𝒅𝑥 = tan 𝑥 - 𝑥 + 𝐶

• ∫ cot²(𝑥) 𝒅𝑥 = -cot 𝑥 - 𝑥 + 𝐶

• General Case for n:

  • ∫ sinⁿ(𝑥) 𝒅𝑥 = -(1/n) sinⁿ⁻¹(𝑥) cos(𝑥) + ((𝑛-1)/𝑛) ∫ sinⁿ⁻²(𝑥) 𝒅𝑥

  • ∫ cosⁿ(𝑥) 𝒅𝑥 = (1/n) cosⁿ⁻¹(𝑥) sin(𝑥) + ((𝑛-1)/𝑛) ∫ cosⁿ⁻²(𝑥) 𝒅𝑥

  • ∫ tanⁿ(𝑥) 𝒅𝑥 = (1/(𝑛-1)) tanⁿ⁻¹(𝑥) - ∫ tanⁿ⁻²(𝑥) 𝒅𝑥 (for n ≠ 1)

Integration of Logs

• Logarithmic Integrals:

  • ∫ logd(𝑎𝑥 + 𝑏) 𝒅𝑥 = (𝑎𝑥 + 𝑏)/𝑎 logd |𝑎𝑥 + 𝑏| + 𝐶

Inverse Trig Functions - Related Formulas

• ∫ 1/(𝑎² + 𝑢²) 𝒅𝑢 = (1/𝑎) tan⁻¹(𝑢/𝑎) + 𝐶 = -(1/𝑎) cot⁻¹(𝑢/𝑎) + 𝐶

• ∫ 1/√(𝑎² - 𝑢²) 𝒅𝑢 = sin⁻¹(𝑢/𝑎) + 𝐶 = -cos⁻¹(𝑢/𝑎) + 𝐶

• ∫ 1/(𝑢 √(𝑢² - 𝑎²)) 𝒅𝑢 = (1/𝑎) sec⁻¹(𝑢/𝑎) + 𝐶 = -(1/𝑎) csc⁻¹(𝑢/𝑎) + 𝐶

Inverse Trigonometric Formulas

• ∫ sin⁻¹(𝑢) 𝒅𝑢 = 𝑢 sin⁻¹(𝑢) + √(1 - 𝑢²) + 𝐶

• ∫ cos⁻¹(𝑢) 𝒅𝑢 = 𝑢 cos⁻¹(𝑢) - √(1 - 𝑢²) + 𝐶

• ∫ tan⁻¹(𝑢) 𝒅𝑢 = 𝑢 tan⁻¹(𝑢) - ln(√(1 + 𝑢²)) + 𝐶

• ∫ cot⁻¹(𝑢) 𝒅𝑢 = 𝑢 cot⁻¹(𝑢) + ln(√(1 + 𝑢²)) + 𝐶

• ∫ sec⁻¹(𝑢) 𝒅𝑢 = 𝑢 sec⁻¹(𝑢) - ln|𝑢 + √(𝑢² - 1)| + 𝐶

• ∫ csc⁻¹(𝑢) 𝒅𝑢 = 𝑢 csc⁻¹(𝑢) + ln|𝑢 + √(𝑢² - 1)| + 𝐶

Table of Integrals

• Formulas for Integrals (u² ± a²):

  • ∫ 1/(𝑢² - 𝑎²) 𝒅𝑢 = (1/2𝑎) ln|𝑢 - 𝑎)/(𝑢 + 𝑎)| + 𝐶 ( |𝑢 - 𝑎| = |𝑎 - 𝑢| )

  • ∫ 1/(𝑎² - 𝑢²) 𝒅𝑢 = (1/2𝑎) ln|𝑢 + 𝑎)/(𝑢 - 𝑎)| + 𝐶

  • ∫ 1/√(𝑢² ± 𝑎²) 𝒅𝑢 = ln|𝑢 + √(𝑢² ± 𝑎²)| + 𝐶

  • ∫ √(𝑢² ± 𝑎²)/𝑢² 𝒅𝑢 = -√(𝑢² ± 𝑎²)/𝑢 + ln|𝑢 + √(𝑢² ± 𝑎²)| + 𝐶

  • ∫ 𝑢²/√(𝑢² ± 𝑎²) 𝒅𝑢 = (1/2)[𝑢√(𝑢² ± 𝑎²) ∓ (𝑎²/2) ln|𝑢 + √(𝑢² ± 𝑎²)|] + 𝐶

  • ∫ 1/(𝑢² √(𝑢² ± 𝑎²) 𝒅𝑢 = ∓(√(𝑢² ± 𝑎²)/(𝑎²𝑢)) + 𝐶

  • ∫ 1/(𝑢² ± 𝑎²)³/² 𝒅𝑢 = ±𝑢/(𝑎² √(𝑢² ± 𝑎²)) + 𝐶

  • ∫ √(𝑎² - 𝑢²) 𝒅𝑢 = −(𝑢/2)√(𝑎² - 𝑢²) + (𝑎²/2) sin⁻¹(𝑢/𝑎) + 𝐶

  • ∫ 1/ℎ + b𝑢 𝒅𝑢 gives the integrals above depending on the choice.