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 – u^{1}v_{1} + u^{11} v_{2} ………

∫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 – u^{1}v_{1} + u^{11} v_{2} – u^{111} v_{3} ………

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.

x^{5} e^{x²}

Solution:

∫x^{5} e^{x²} dx = ∫x x^{4} e^{x²} dx

Let t = x²

\(\frac { dt }{dx}\) = 2x

dt = 2xdx

xdx = \(\frac { dt }{2}\)

∫x x^{4} e^{x²} dx = ∫t² e^{t} (\(\frac { dt }{2}\))