Integration

Question 1

What must we always remember for integration without limits

Answer 1

To add C (+C)

Question 2

Basic rule for integration

Answer 2

Raise power by 1 and divide by new power

Question 3

Chain rule for integration

Answer 3

When function with function we divide by the derivative of the function within. Or multiply by 1/derivative of function within.

Question 4

§sinxdx=

Answer 4

-cosx +c

Question 5

§cosxdx=

Answer 5

sinx +c

Question 6

§e^xdx=

Answer 6

e^x +c

Question 7

§e^3xdx=

Answer 7

1/3e^3x +c

Question 8

What can we do to make integrals easier to evaluate

Answer 8

Take constants out the integral and deal with them after.

Question 9

§1/3x+4dx

Answer 9

1/3 ln|3x+4| +c

Question 10

Special rule for integration

Answer 10

§f’(x)/f(x)dx= ln(f(x) + c

Question 11

What to look out for when doing integration involving a fraction

Answer 11

Special rule for integration, if top line is or can be derivative of top line

Question 12

What can we do to help special rule for integration be the case

Answer 12

Manipulate the fraction then take multipliers outside the integral. Or add another fraction that makes the special rule work for one, and is still equal to same equation.

Question 13

When integrating partial fractions for a Bx+C what should we do

Answer 13

Manipulate to use special rule for integration

Question 14

Integration by substitution basic processes

Answer 14

Write out full integral.
For our u =f(x) we find du/dx and then write on one line, one side du=f’(x)dx we substitute in our initial value for u and also a du or 1/f’(x) du. Simplify our equation. Integrate. At the end re substitute our original variables.

Question 15

What to remember for integration with definite integrals

Answer 15

We must change the definite integrals, we do this by subbing our definite integrals into u=x equation to work our u.
We use these in our square brackets to get our answer at the end.

Question 16

Equation for integration by parts:

Answer 16

§uv’=uv-§u’v

Question 17

In integration by parts u should ….

Answer 17

Get more simple and eventually differentiate to a constant.

Question 18

Process of integration by parts.

Answer 18

Set up integral and evaluate which value should be u and which should be v’.
At side of page write down all values of u,v,u’ and v’. (We differentiate u and integrate v’).
Use our formula and follow it to conclusion. Remember to +c.

Question 19

Integration by parts twice

Answer 19

If second integral is not simple then we may have to integrate again. Do same process, remember to bracket as it is being taken away.

Question 20

What to do when integrating by parts with cyclic functions eg. (e^x and sin^x)

Answer 20

Set original integral = I.
Carry out integration by parts (usually twice), we may have to take a constant out of the integral. Substitute in I for integral. Move I to LHS so it’s a multiple I. Divide by multiple of I on LHS on both sides. Change c to C