Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.7 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.7

Integrate the following with respect to x:

Question 1.
(i) 9x e3x
Answer:

(ii) x sin 3x
Answer:

(iii) 25 x e-5x
Answer:

(iv) x sec x tan x
Answer:
∫ x sec x tan x dx
u = x, u’ = 1 , u” = 0
dv = sec x tan x dx , v = ∫ sec x tan x dx = sec x
v1 = ∫ v dx = ∫ sec x dx = log |sec x + tan x|
v2 = ∫ v1 dx = ∫ log |sec x + tan x| dx
∫ u dv = uv – u’ v1 + u” v2 – u”’ v3 + ………….
∫x sec x tan x dx = x sec x – 1 × log |sec x + tan x| + 0 × ∫ log |x sec x tan x| + c
∫x sec x tan x dx = x sec x – log |sec x + tan x| + c

Question 2.
(i) x log x
Answer:

(ii) 27 x2 e3x
Answer:

(iii) x2 cos x
Answer:

(iv) x3 sin x
Answer:
∫ x3 sin x
u = x3, u’ = 3x2, u” = 6x , u”’ = 6 , u’v = 0
dv = sin x dx ⇒ v = ∫ sin x dx = – cosx
v1 = ∫v dx = ∫- cos x dx = – ∫ cos x dx = – sin x
v2 = ∫ v1 dx = ∫ – sin x dx = – ∫ sin x dx = – (- cos x) = cos x
v3 = ∫ v2 dx = ∫ cos x dx = sin x
v4 = ∫ v3 dx = ∫ sin x dx = – cos x
∫ u dv = uv – u’ v1 + u” v2 – u”’ v3 + u’vv4 – ………..
∫x3 sin x dx = x3 (- cos x) – 3x2 (- sin x) + 6x (cos x) – 6 sin x + 0 (- cos x) + c
= -x3 cos x + 3x2 sin x + 6x cos x – 6 sin x + c

Question 3.
(i)
Answer:

(ii) x5 ex2
Answer:

(iii)
Answer:

(iv)
Answer: