# Basic Integration Formulae

```COS-MATH-231
Spring 2019
Basic Integration Formulas
β« πππ₯ = ππ₯ + πΆ
β« π₯ π ππ₯ =
β«
(any number π)
π₯ π+1
+ πΆ,
π+1
β«
β«
(π β  β1)
ππ₯
= ln|π₯| + πΆ
π₯
β« π ππ₯ ππ₯ =
β«
1 ππ₯
π + πΆ,
π
β«
( π β  0)
1
β« πππ₯ ππ₯ = (
) πππ₯ + πΆ,
π ln π
(π β  0 and π > 0,
β«
ππ₯
1
= ln|ππ₯ + π| + πΆ,
ππ₯ + π π
β«
ππ₯
β1
1
=
+ πΆ,
π
(ππ₯ + π)
π(π β 1) (ππ₯ + π)πβ1
1
sin ππ₯ + πΆ,
π
β« sec 2 ππ₯ ππ₯ =
1
tan ππ₯ + πΆ,
π
β« sec ππ₯ tan ππ₯ ππ₯ =
πβ 0
π₯2
ππ₯
1
π₯
= tanβ1 + πΆ,
2
+π
π
π
πβ 0
ππ₯
βπ2
π₯2
β
ππ₯
π₯
= sinβ1 ( ) + πΆ,
π
π₯ βπ₯ 2 β π2
=
1
π₯
sec β1 | | + πΆ,
π
π
β« ln π₯ ππ₯ = π₯ ln π₯ β π₯ + πΆ
β« cot π₯ ππ₯ = ln|sin π₯| + πΆ
β« sec π₯ ππ₯ = ln|sec π₯ + tan π₯| + πΆ
πβ 0
πβ 0
πβ 0
1
β« csc ππ₯ cot ππ₯ ππ₯ = β csc ππ₯ + πΆ,
π
( π β  0 and π β  1)
β« tan π₯ ππ₯ = ln|sec π₯| + πΆ
πβ 0
1
sec ππ₯ + πΆ,
π
1
β« csc 2 ππ₯ ππ₯ = β cot ππ₯ + πΆ,
π
ππ₯
1
π₯βπ
=
ln |
| + πΆ,
2
βπ
2π π₯ + π
(π β  0)
1
β« sin ππ₯ ππ₯ = β cos ππ₯ + πΆ, π β  0
π
β« cos ππ₯ ππ₯ =
π β  1)
π₯2
πβ 0
β« csc π₯ ππ₯ = βln|csc π₯ + cot π₯| + πΆ
β« tan2 π₯ ππ₯ = tan π₯ β π₯ + πΆ
πβ 0
πβ 0
```