Samacheer Kalvi 12th Business Maths Guide Chapter 2 Integral Calculus I Ex 2.5

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 2 Integral Calculus I Ex 2.5 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Ex 2.5

Question 1.
Integrate the following with respect to x.
xe^-x
Solution:
= ∫xe^-x dx = ∫udv
∫udv = uv – u¹v₁ + u¹¹ v₂ ………
∫xe^-x dx = (x) (-e^-x) – (1) e^-x + c
= -xe^-x – e^-x + c
= -e^-x (x + 1) + c

Question 2.
x³e^3x
Solution:
= ∫x³e^3x dx = ∫udv
∫udv = uv – u¹v₁ + u¹¹ v₂ – u¹¹¹ v₃ ………

Question 3.
log x
Solution:
∫log x dx = ∫udv
∫udv = uv – ∫vdu
∫log x dx = (log x) (x) – ∫(x) (\(\frac { dx }{x}\)) + c
= x log x – ∫dx + c
= x log x – x + c
= x(log x – 1) + c

Question 4.
x log x
Solution:
∫x log x dx = ∫udv
∫udv = uv – ∫vdu

Question 5.
\(\sqrt { 1-sin 2x }\)
Solution:
xⁿ log x
∫xⁿ log x dx = ∫udv
∫udv = uv – ∫vdu

Question 6.
x⁵ e^x²
Solution:
∫x⁵ e^x² dx = ∫x x⁴ e^x² dx
Let t = x²
\(\frac { dt }{dx}\) = 2x
dt = 2xdx
xdx = \(\frac { dt }{2}\)
∫x x⁴ e^x² dx = ∫t² e^t (\(\frac { dt }{2}\))