Integration

Question 1

Integration by parts

Answer 1

∫ u dv = uv – ∫ v du

Question 2

∫ lnx dx

Answer 2

xlnx – x + C

Question 3

∫ sinx dx

Answer 3

–cosx + C

Question 4

∫ cosx dx

Answer 4

sinx + C

Question 5

∫ tanx dx

Answer 5

–ln |cosx| + C

Question 6

∫ cotx dx

Answer 6

ln |sinx| + C

Question 7

∫ secx dx

Answer 7

ln |secx + tanx| + C

Question 8

∫ cscx dx

Answer 8

–ln |cscx + cotx| + C

Question 9

∫ sin^-1x dx

Answer 9

x sin^-1x + √(1 – x²) + C

Question 10

∫ cos^-1x dx

Answer 10

x cos^-1x – √(1 – x²) + C

Question 11

∫ tan^-1x dx

Answer 11

x tan^-1x – ¹/₂ ln |x² + 1| + C

Question 12

∫ cot^-1x dx

Answer 12

x cot^-1x + ¹/₂ ln |x² + 1| + C

Question 13

∫ sec^-1x dx

Answer 13

x sec^-1x – sgnx ln |x + √(x² – 1)| + C

Question 14

∫ csc^-1x dx

Answer 14

x csc^-1x + sgnx ln |x + √(x² – 1)| + C

Question 15

∫_a^∞ ƒ(x) dx

Answer 15

lim_b→∞ ∫_a^b ƒ(x) dx

Question 16

∫_-∞^b ƒ(x) dx

Answer 16

lim_a→-∞ ∫_a^b ƒ(x) dx

Question 17

∫_a^b ƒ(x) dx =

If ƒ is discontinuous at x = c in [a, b]

Answer 17

lim_k→c⁺ ∫_k^b ƒ(x) dx + lim_k→c^– ∫_a^k ƒ(x) dx

Question 18

∫ du / √ ( 1 – u² )

Answer 18

sin^-1(u) + C

for u² < 1

Question 19

∫ du / ( 1 + u² )

Answer 19

tan^-1(u) + C

Question 20

∫ du / ( u √( u² – 1) )

Answer 20

sec^-1(u) + C

for u² > 1

Question 21

∫a^x dx

Answer 21

1 / lna a^x + C