Integrals in depth

Question 1

Integral of tan (u)

Answer 1

• ln |cos(x)|+ c

Question 2

Integral of cot (u)

Answer 2

ln |sin(x)| + c

Question 3

Integral of sec (u)

Answer 3

ln |sec(x) + tan(x)| + c

Question 4

Integral of csc (u)

Answer 4

• ln |csc(x) + cot(x)| + c

Question 5

Integration by parts

Answer 5

( v * u ) - integral (v * du)
How to choose U:
(BEST) Logs , inverse trig , algebra , trig , exponential (WORST)

Question 6

Table method

Answer 6

Use when integration by parts is repeated and/or derivative reaches zero after some time

Question 7

If integration cycles back to original (integration by parts)

Answer 7

Once cycled back to original, set the term “I” as the common term cycled back.
Place an “I” on the left side of the equation and then move terms with “I” on the right side to the left side.
Once “I” is separated, get “I” by itself without any constants.

Question 8

Partial fractions

Answer 8

1) degree of numerator must be less than denominator
2) factor denominator
3) set up fraction decomp
–> If the factor is linear (x + 1), numerator will be a constant (ex: A)
–> If the factor is an irreducible quadratic (x² + 1), numerator will be a linear expression (ex: Bx + C)
–> If a factor is repeated ((x - 2)³), create fractions with each power of that factor as the denominator.
4) either plug in x value or make coefficients the same
5) substitute the values back into the partial fractions