Calculus 2 (INTEGRAL CALCULUS)
Module 3
Integrals Leading to Logarithms
TEACHER:
Engr. Herdinio Z. Caneja
Bachelor of Science in Electrical Engineering ( B.S.E.E )
1.
The Basic Logarithmic Form Formula:
𝒅𝒖
𝒖
= 𝐥𝐧 𝒖 + 𝑪
Examples1 :
then du= 4x³ dx
We lack of a constant 4 in the derivative du, so introduce 4 inside the integral
and neutralize it outside the integral by 1/4, then
= 2. 1/4 4𝑥 3 𝑑𝑥/(𝑥 4 + 1) , using the formula :
NOTE: We can only neutralize a constant and never a variable.
Finally,
= ½ .ln (x⁴+1) + C
2.
using the formula :
Then: = ln (4+ tanx) + C
3.
We lack of a constant 2 in the derivative du, so introduce 2 inside the integral and
neutralize it outside the integral by 1/2, then using the formula :
= ½ . ln (1+2.lnx) + C
4.
We lack of a constant 3 in the derivative du, so introduce 3 inside the integral and
neutralize it outside the integral by 1/3,
Substituting to the formula,
= 1/3. ln (x³ + 3x) + C
5. ( 𝑥 3 +𝑥 − 3) 𝑑𝑥 / (𝑥 − 1)
Special NOTE:
If the power of a polynomials in x at the numerator is greater than the power
of a polynomials in x at the denominator, always perform division until such time
that the power of a polynomials in x at the numerator is already less than the
power of a polynomials in x at the denominator.
So, dividing the numerator by the denominator, we have
(x³ + x – 3)/(x-1) = x² +x + 2 – 1/(x-1)
Thus ,
= [ 𝑥 2 + 𝑥 + 2 − 1/(𝑥 − 1)] .dx
= 𝑥 2 + 𝑥 + 2 𝑑𝑥 - 𝑑𝑥/(𝑥 − 1)
= 𝑥 2 𝑑𝑥 + 𝑥𝑑𝑥 + 2 𝑑𝑥 - 𝑑𝑥/(𝑥 − 1)
For: 𝑑𝑥/(𝑥 − 1) ; u=(x-1) and du=dx (satisfied)
Then finally,
= x³/3 + x²/2 + 2x – ln(x-1) + C
Assessment for Module 3 – Integrals leading to Logarithms :
Evaluate each integral and check by differentiation:
1.
2𝑑𝑥
1−5𝑥
= ____________________
9.
2.
𝑥−1 𝑑𝑥
(𝑥 2 −2𝑥+16)
= ____________________
10.
3.
4.
𝑡𝑑𝑡
1+𝑡 2 3
𝑥 2 +6 𝑑𝑥
= _______________________
11.
𝑑𝑥
𝑥𝑙𝑛𝑥
= _________________________
𝑐𝑠𝑐𝑥 . 𝑐𝑜𝑡𝑥 𝑑𝑥
3+2𝑐𝑠𝑐𝑥
𝑒 3𝑡 𝑑𝑡
1+𝑒 3𝑡
= ___________________
= _________________________
𝑒 2𝑥 2𝑥.𝑙𝑛𝑥+1 𝑑𝑥
𝑥(1+𝑒 2𝑥 .𝑙𝑛𝑥)
= _________________________
12.
5.
tan 3ϴ dϴ = __________________________________
13.
𝑐𝑠𝑐∅. 𝑑∅ = ___________________________________
6.
(sec 2 ∅)𝑑∅. 1+𝑡𝑎𝑛∅ = ___________________________
14.
𝑑∅/𝑐𝑜𝑠2∅ = _________________________________
15.
sec 2 2∅. 𝑑∅. 𝑡𝑎𝑛2∅ + 3 = _________________________
16.
𝑥 4 𝑑𝑥
𝑥+4
7.
8.
𝑥
1
𝑡 3 +3𝑡 𝑑𝑡
𝑡 4 +6𝑡 2 +3
= _____________________________________
𝑥 3 +2𝑥−4 𝑑𝑥
𝑥+1
= _________________________________
= _____________________
1
= __________________________________________
17.
𝑥 3 −6 𝑑𝑥
𝑥 4 −24𝑥+3 ⅗
18.
𝑑𝑥/(
19.
𝑒 3𝑥 − 𝑒 −3𝑥
𝑒 3𝑥 + 𝑒 −3𝑥
20.
4𝑑𝑥
3+𝑥
= ______________________________
𝑥+1
3
= ___________________________
𝑑𝑥 = _____________________________
= ____________________________________