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Integration Formulas
Integration Formulas
The Power Rule
Basic Integration Formulas:
β« π π π = ππ₯ + π
β« πβΏ π π = π₯βΏβΊΒΉ / (π + 1) + πΆ (where n β -1)
β«(ππ + π)βΏ π π = 1/π β (ππ₯ + π)βΏβΊΒΉ / (π + 1) + πΆ
Rational Functions
Integration of Rational Functions:
β« 1/π₯ π π = ln |π₯| + πΆ
β« 1/(ππ₯ + π) π π = 1/π ln |ππ₯ + π| + πΆ
Exponential Functions
Integration of Exponential Functions:
β« eΛ‘Λ£ π π = eΛ‘Λ£ + πΆ
β« e^(ππ₯ + π) π π = 1/π e^(ππ₯ + π) + πΆ
β« π^ππ₯ + π π π = πππ₯ + π/π ln π + πΆ
Trigonometric Functions
Integration of Trigonometric Functions:
β« sin π₯ π π = -cos π₯ + πΆ
β« sin(ππ₯ + π) π π = -1/π cos(ππ₯ + π) + πΆ
β« cos π₯ π π = sin π₯ + πΆ
β« cos(ππ₯ + π) π π = 1/π sin(ππ₯ + π) + πΆ
β« csc π₯ cot π₯ π π = -csc π₯ + πΆ
β« sec π₯ tan π₯ π π = sec π₯ + πΆ
β« tan π₯ π π = ln |sec π₯| + πΆ = -ln |cos π₯| + πΆ
β« cot π₯ π π = ln |sin π₯| + πΆ = -ln |csc π₯| + πΆ
β« sec π₯ π π = ln |sec π₯ + tan π₯| + πΆ
β« csc π₯ π π = ln |csc π₯ - cot π₯| + πΆ
Integration By Parts
Formula for Integration by Parts:
β« π π π = π’π£ - β« π£ π π’
Extended case: β« ππ π π = π’π£π€ - β« π£π€ π π’ - β« ππ€ π π£
Integration By Parts - Related Formulas
β« e^(ππ₯) sin(ππ₯) π π = (e^(ππ₯)/(πΒ² + πΒ²)) [π sin(ππ₯) - π cos(ππ₯)] + πΆ
β« e^(ππ₯) cos(ππ₯) π π = (e^(ππ₯)/(πΒ² + πΒ²)) [π cos(ππ₯) + π sin(ππ₯)] + πΆ
β« π₯βΏ ln(π₯) π π = (π₯βΏβΊΒΉ / (π + 1)) ln(π₯) - (π₯βΏβΊΒΉ / (π + 1)Β²) + πΆ (π β 1)
β« π₯βΏ e^(ππ₯) π π = (π₯βΏ e^(ππ₯)/π) - (π/π) β« π₯βΏβ»ΒΉ e^(ππ₯) π π₯
Trig Substitution
**For Trig Substitution: **
β« β(πΒ² - π’Β²) π π’ = (π’/2) β(πΒ² - π’Β²) + (πΒ²/2) sinβ»ΒΉ(π’/π) + πΆSubstitution: π’ = π sin(π) , π π’ = π cos(π) π π
β« β(πΒ² + π’Β²) π π’ = (π’/2) β(π’Β² + πΒ²) + (πΒ²/2) ln |π’ + β(π’Β² + πΒ²)| + πΆSubstitution: π’ = π tan(π) , π π’ = π secΒ²(π) π π
β« β(π’Β² - πΒ²) π π’ = (π’/2) β(π’Β² - πΒ²) - (πΒ²/2) ln |π’ + β(π’Β² - πΒ²)| + πΆSubstitution: π’ = π sec(π) , π π’ = π sec(π) tan(π) π π
Reduction Formulas
β« sinΒ²(π₯) π π₯ = (1/2) π₯ - (1/4) sin(2π₯) + πΆ = (1/2)(π₯ - sin π₯ cos π₯) + πΆ
β« cosΒ²(π₯) π π₯ = (1/2) π₯ + (1/4) sin(2π₯) + πΆ = (1/2)(π₯ + sin π₯ cos π₯) + πΆ
β« tanΒ²(π₯) π π₯ = tan π₯ - π₯ + πΆ
β« cotΒ²(π₯) π π₯ = -cot π₯ - π₯ + πΆ
General Case for n:
β« sinβΏ(π₯) π π₯ = -(1/n) sinβΏβ»ΒΉ(π₯) cos(π₯) + ((π-1)/π) β« sinβΏβ»Β²(π₯) π π₯
β« cosβΏ(π₯) π π₯ = (1/n) cosβΏβ»ΒΉ(π₯) sin(π₯) + ((π-1)/π) β« cosβΏβ»Β²(π₯) π π₯
β« tanβΏ(π₯) π π₯ = (1/(π-1)) tanβΏβ»ΒΉ(π₯) - β« tanβΏβ»Β²(π₯) π π₯ (for n β 1)
Integration of Logs
Logarithmic Integrals:
β« logd(ππ₯ + π) π π₯ = (ππ₯ + π)/π logd |ππ₯ + π| + πΆ
Inverse Trig Functions - Related Formulas
β« 1/(πΒ² + π’Β²) π π’ = (1/π) tanβ»ΒΉ(π’/π) + πΆ = -(1/π) cotβ»ΒΉ(π’/π) + πΆ
β« 1/β(πΒ² - π’Β²) π π’ = sinβ»ΒΉ(π’/π) + πΆ = -cosβ»ΒΉ(π’/π) + πΆ
β« 1/(π’ β(π’Β² - πΒ²)) π π’ = (1/π) secβ»ΒΉ(π’/π) + πΆ = -(1/π) cscβ»ΒΉ(π’/π) + πΆ
Inverse Trigonometric Formulas
β« sinβ»ΒΉ(π’) π π’ = π’ sinβ»ΒΉ(π’) + β(1 - π’Β²) + πΆ
β« cosβ»ΒΉ(π’) π π’ = π’ cosβ»ΒΉ(π’) - β(1 - π’Β²) + πΆ
β« tanβ»ΒΉ(π’) π π’ = π’ tanβ»ΒΉ(π’) - ln(β(1 + π’Β²)) + πΆ
β« cotβ»ΒΉ(π’) π π’ = π’ cotβ»ΒΉ(π’) + ln(β(1 + π’Β²)) + πΆ
β« secβ»ΒΉ(π’) π π’ = π’ secβ»ΒΉ(π’) - ln|π’ + β(π’Β² - 1)| + πΆ
β« cscβ»ΒΉ(π’) π π’ = π’ cscβ»ΒΉ(π’) + ln|π’ + β(π’Β² - 1)| + πΆ
Table of Integrals
Formulas for Integrals (uΒ² Β± aΒ²):
β« 1/(π’Β² - πΒ²) π π’ = (1/2π) ln|π’ - π)/(π’ + π)| + πΆ ( |π’ - π| = |π - π’| )
β« 1/(πΒ² - π’Β²) π π’ = (1/2π) ln|π’ + π)/(π’ - π)| + πΆ
β« 1/β(π’Β² Β± πΒ²) π π’ = ln|π’ + β(π’Β² Β± πΒ²)| + πΆ
β« β(π’Β² Β± πΒ²)/π’Β² π π’ = -β(π’Β² Β± πΒ²)/π’ + ln|π’ + β(π’Β² Β± πΒ²)| + πΆ
β« π’Β²/β(π’Β² Β± πΒ²) π π’ = (1/2)[π’β(π’Β² Β± πΒ²) β (πΒ²/2) ln|π’ + β(π’Β² Β± πΒ²)|] + πΆ
β« 1/(π’Β² β(π’Β² Β± πΒ²) π π’ = β(β(π’Β² Β± πΒ²)/(πΒ²π’)) + πΆ
β« 1/(π’Β² Β± πΒ²)Β³/Β² π π’ = Β±π’/(πΒ² β(π’Β² Β± πΒ²)) + πΆ
β« β(πΒ² - π’Β²) π π’ = β(π’/2)β(πΒ² - π’Β²) + (πΒ²/2) sinβ»ΒΉ(π’/π) + πΆ
β« 1/β + bπ’ π π’ gives the integrals above depending on the choice.
NoteStudied by 5 peopleNoteStudied by 19 peopleNoteStudied by 7 peopleNoteStudied by 18 peopleNoteStudied by 6628 peopleNoteStudied by 55 people