Integrals

Question 1

The Antiderivative

Answer 1

A derivative in reverse.

The antiderviatve of ƒ(x) is F(x).

Every function has an infinte number of antiderivatives (+C).

When you take the intgral (or anitderivative) of a function of x, you always add the term dx to the integand. (or dy if it’s a function of y, etc.)

Question 2

U-Substitution

Answer 2

In its most basic form, u-substitution is used when an integral contains some function and its derivative, that is, for an integral of the form ∫(f(x)f’(x)dx). The integration is achieved by rewriting the integral in a form that makes it easier to read. Here, let u=f(x). Then du/dx=f’(x), so, we can say du=f’(x)dx and the integral becomes ∫(udu). This integral can now be easily evaluated; we know ∫(udu)=(u²)/2 + C . The substitution is then reversed, giving us (f(x))²/2+ C.

Question 3

Integration by parts

∫u dv =

Answer 3

uv – ∫v du

Question 4

∫k du

Answer 4

ku + C

Question 5

∫uⁿ du

Answer 5

uⁿ⁺¹ / (n+1) + C

Question 6

∫ du/u

Answer 6

ln |u| + C

Question 7

∫e^u du

Answer 7

e^u + C

Question 8

∫lnu du

Answer 8

u lnu – u + C

Question 9

∫a^u du

Answer 9

1/ lna • a^u + C

Question 10

∫sin(u) du

Answer 10

-cos(u) + C

Question 11

∫cos(u) du

Answer 11

sin(u) + C

Question 12

∫tan(u) du

Answer 12

-ln|cosu| + C

ln|secu| + C

Question 13

∫cot(u) du

Answer 13

ln|sinu| + C

Question 14

∫sec(u) du

Answer 14

ln|secu + tanu| + C

Question 15

∫csc(u) du

Answer 15

-ln|cscu + cotu| + C

Question 16

∫sec²(u) du

Answer 16

tan(u) + C

Question 17

∫csc²(u) du

Answer 17

-cot(u) + C

Question 18

∫ sec(u) • tan(u) du

Answer 18

sec(u) + C

Question 19

∫ csc(u) • cot(u) du

Answer 19

-csc(u) + C

Question 20

∫ du / √ ( 1 – u² )

Answer 20

sin^-1(u) + C

for u² < 1

Question 21

∫ du / ( 1 + u² )

Answer 21

tan^-1(u) + C

Question 22

∫ du / ( u √( u² – 1) )

Answer 22

sec^-1(u) + C

Question 23

∫ sinⁿ(u) • cos(u) du

Answer 23

sinⁿ⁺¹(u) / _n+1 + C