U3 - Integration

Question 1

Integration and differentiation are…

Answer 1

Inverse operations.

Derivative of F(x) is f(x). Anti-derivative of f(x) is F(x).

Question 2

What is an indefinite integral (F f(x)dx )?

Answer 2

The integral of f with respect to x’.

If F’(x)=f(x), then F f(x)dx=F(x)+c

Question 3

Rules for INDEFINITE integrals:

Answer 3

Power rule:
F x^n dx = [1/(n+1)]^n+1 +C (n≠1)

Constant:
F k dx=kx + C

Constant multiple: F kf(x)dx=k F f(x)dx

Sum or difference:
F [f(x)+/-g(x)dx]=F f(x)dx +/- F g(x)dx

Question 4

What are the SPECIAL FUNCTION RULES for integration?

Answer 4

Euler’s:
F e^x dx=e^x + C

x as denom:
F 1/x dx=ln |x| + C

Cos:
F cos(x)dx=sin(x) + C

Sin:
F sin(x)dx=-cos(x) + C

Sec^2:
F [sec(x)]^2dx=tan(x) + C

Question 5

What are the rules for integrating f(ax+b) and INTEGRATION BY SUBSTITUTION?

Answer 5

Euler’s:
F e^(ax+b) dx=(1/a)e^(ax+b) + C (a≠0)

X as denom:
F 1/(ax+b) dx=(1/a) ln |ax+b| + C (a≠0)

F (ax+b)^n dx=(1/a) [(ax+b)^n+1]/(n+1) + C