Samacheer Kalvi 12th Business Maths Guide Chapter 2 Integral Calculus I Ex 2.4

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 2 Integral Calculus I Ex 2.4 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Ex 2.4

Question 1.
Integrate the following with respect to x.
2 cos x – 3 sin x + 4 sec² x – 5 cosec² x
Solution:
∫2 cos x – 3 sin x + 4 sec² x – 5 cosec² x
= 2 ∫ cos x dx – 3 ∫ sin x dx + 4 ∫ sec² x – 5 ∫ cosec² x dx
= 2 sin x + 3 cos x + 4 tan x + 5 cot x + c

Question 2.
sin³x
Solution:
sin³x = 3 sin x – 4sin³x
4sin³x = 3 sin x – sin3x
⇒ sin³x = \(\frac { 1 }{4}\) [3 sin x – sin 3x]
∫sin³x dx = \(\frac { 1 }{4}\) ∫(3 sin x – sin3x) dx
= \(\frac { 1 }{4}\) [3(-cos x) – (\(\frac { -cos3x }{2}\)) + c
= \(\frac { -3 }{4}\) [cos x] + \(\frac { 1 }{12}\) (cos 3x) + c

Question 3.
\(\frac { cos 2x+2sin^2x }{cos^2x}\)
Solution:

Question 4.
\(\frac { 1} {sin^2x cos^2x}\)
Solution:

Question 5.
\(\sqrt { 1-sin 2x }\)
Solution:
\(\sqrt { 1-sin 2x }\) dx = ∫\(\sqrt { sin^2x+cos^2x-2sinx cos x dx }\)
∫\(\sqrt { (sin x-cos x)^2 }\) dx
∫(sin x – cos x) dx = ∫sin x dx – ∫cos x dx
= -cos x – sin x + c
= -[cos x + sin x] + c