Integrate with respect to x :
(i) 
(ii) 
(iii) 
(iv) 
Text Solution
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(i) n 
+ C
(ii) 
n |x + 3| – 
n |x 2 + 1| + 
tan –1 x + C
(iii) 4 n|x + 1| + 
– 4 n|x + 2| + C
(iv) 
n |x + 1| – n |x + 2| + 
n |x + 3| + C
Sol. (i) 
= 
= n 
+ C
(ii) Ι = 
Let 
= 
⇒ A (x 2 +1) + (Bx + C)(x + 3) = 1
Put x = –3 ⇒ A = 1/10
and A + 3C = 1 A + B = 0
B = –1/10
C = 3/10
⇒ Ι = 
= 
n |x + 3| – 
n |x 2 + 1| + 
tan –1 x + K
(iii) Ι = 
= 
+ 
+ 
= 4 
– 
– 4 
= 4 n|x + 1| + 
– 4 n|x + 2| + C
(iv) Ι = 
put x + 1 = t ⇒ dx = dt
⇒ Ι = 
= 
= 
= 
= 
n|t| – n|t +1| + 
n |t + 2| + C
= 
n |x + 1| – n |x + 2| + 
n |x + 3| + C
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