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.

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## 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.
x³e3x
Solution:
= ∫x³e3x 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
Solution:
∫x5 e dx = ∫x x4 e dx
Let t = x²
$$\frac { dt }{dx}$$ = 2x
dt = 2xdx
xdx = $$\frac { dt }{2}$$
∫x x4 e dx = ∫t² et ($$\frac { dt }{2}$$)