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