MATHEMATICS – Integral Calculus 1
Vernante 11
INTEGRATION FORMULAS
Algebraic, Exponential, & Logarithmic Functions
1. ∫𝑑𝑢 = 𝑢 + 𝐶
2. ∫𝑎 𝑑𝑢 = 𝑎 ∫𝑑𝑢 =𝑎𝑢 + 𝐶
3. ∫𝑎𝑢𝑑𝑢 =𝑎𝑢
ln 𝑎+ 𝐶 , 𝑎 > 1, 𝑎 ≠ 1
4. ∫𝑢𝑛𝑑𝑢 =𝑢𝑛+1
𝑛+1 + 𝐶 for 𝑛 ≠ −1
5. ∫𝑒𝑢𝑑𝑢 = 𝑒𝑢+ 𝐶
6. ∫𝑢−1𝑑𝑢 =∫𝑑𝑢
𝑢=ln|𝑢|+ 𝐶
7. ∫ln|𝑢|𝑑𝑢 = 𝑢 ln|𝑢|− 𝑢 + 𝐶
Trigonometric Functions
8. ∫sin 𝑢𝑑𝑢 = − cos 𝑢 + 𝐶
9. ∫cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶
10. ∫tan 𝑢 𝑑𝑢 =ln|sec 𝑢|+ 𝐶
11. ∫cot 𝑢 𝑑𝑢 =ln|sin 𝑢|+ 𝐶
12. ∫sec 𝑢 𝑑𝑢 =ln|sec 𝑢 + tan 𝑢|+ 𝐶
13. ∫csc 𝑢𝑑𝑢 =ln|csc 𝑢 − cot 𝑢|+ 𝐶
14. ∫sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶
15. ∫csc 𝑢 cot 𝑢 𝑑𝑢 = −csc 𝑢 + 𝐶
16. ∫sec2𝑢𝑑𝑢 = tan 𝑢 + 𝐶
17. ∫csc2𝑢𝑑𝑢 = −cot 𝑢 + 𝐶
Inverse Trigonometric Functions
18. ∫du
√𝑎2−𝑢2= arcsin u
a+ C
19. ∫du
𝑎2+𝑢2=1
aarctan u
a+ C
20. ∫du
𝑢√𝑢2−𝑎2=1
aarcsec u
a+ C
21. ∫arcsin 𝑢 𝑑𝑢 = 𝑢 arcsin 𝑢 + √1 − 𝑢2+ 𝐶
22. ∫arctan 𝑢 𝑑𝑢 = 𝑢 arctan 𝑢 − ln √1 + 𝑢2+ 𝐶
Integration by Parts
23. ∫udv = uv −∫vdu
SAMPLE PROBLEMS
1. Given 𝑓(𝑥)=∫(3𝑥3− 7𝑥) 𝑑𝑥, and it is known that
𝑓(𝑥)= 4 when 𝑥 = 2. What is the value of 𝑓(𝑥)
when 𝑥 = 0?
a. 2 c. 0
b. 8 d. 6
2. Evaluate ∫𝑥
𝑥2+2 𝑑𝑥
a. ln √𝑥2+ 2 + 𝐶 c. ln √𝑥2+ 2
b. ln(𝑥2+ 2)+ 𝐶 d. ln(𝑥2+ 2)
3. Evaluate ∫𝑥cos(2𝑥2+ 7) 𝑑𝑥
a. 1
4sin(2𝑥2+ 7)+ 𝐶 c. 1
4cos(2𝑥2+ 7)+ 𝐶
b. sin(2𝑥2+ 7)+ 𝐶 d. cos(2𝑥2+ 7)+ 𝐶
4. Evaluate ∫𝑒𝑥𝑑𝑥
1+𝑒2𝑥
a. 1
2ln(1 + 𝑒2𝑥 )+ 𝐶 c. 1
2arctan(𝑒𝑥)+ 𝐶
b. ln(1 + 𝑒2𝑥 )+ 𝐶 d. arctan(𝑒𝑥)+ 𝐶
5. Evaluate ∫𝑒𝑥sin 𝑥𝑑𝑥
a. −𝑒𝑥(sin 𝑥 − cos 𝑥 )+ 𝐶
b. 𝑒𝑥
2(cos 𝑥 + sin 𝑥)+ 𝐶
c. 𝑒𝑥
2(sin 𝑥 − cos 𝑥)+ 𝐶
d. −𝑒𝑥(cos 𝑥 + sin 𝑥)+ 𝐶
6. The expression ∫𝑥
√9−𝑥2 𝑑𝑥 is equal to _____________.
a. 2√9 − 𝑥2+ 𝐶 c. −1
4√9 − 𝑥2+ 𝐶
b. −1
2ln √9 − 𝑥2+ 𝐶 d. −√9 − 𝑥2+ 𝐶
7. The integral of ∫sin 2θ 𝑑θ
𝜋
𝜋/4 is equal to ___________.
a. 2 c. -1/2
b. -2 d. 1/2
8. The expression ∫𝑥𝑒𝑥
1
0𝑑𝑥 is equal to _____________.
a. 2 − 𝑒 c. 1
b. -1 d. 𝑒 − 1
9. The integral of ∫sin3θ cos θ dθ
𝜋/2
𝜋/4 is equal to _____.
a. -3/16 c. 1/8
b. 3/16 d. -1/8
10. __________ is a definite integral which is not equal to
zero.
a. ∫cos 𝑥 𝑑𝑥
𝜋
0 c. ∫cos3𝑥𝑑𝑥
𝜋
−𝜋
b. ∫cos2𝑥𝑑𝑥
𝜋
−𝜋 d. ∫𝑥2sin 𝑥𝑑𝑥
𝜋
−𝜋
11. The average value of 𝑔(𝑥)=(𝑥 − 3)2 in the
interval (1, 3) is ________.
a. 2/3 c. 2
b. 8/3 d. 4/3
