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Use the substitute x= tan y to show that

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2 Answers

x = tany => x = 0, y = 0; x = 1, y = pi/4
dx = sec2y dy
∫{0,1}1/(1+x2)2 dx
= ∫{0, pi/4} cos2y dy
= ∫{0, pi/4} (1/2)(1+cos2y) dy
= (1/2)y + (1/4)sin(2y) from 0 to pi/4
= pi/8 + 1/4, since sin(2*pi/4) = 1.
Use identity 1/(1+tan(x)2)=cos (x)2 Your integral becomes, upon substitution x=tan (y),
01 dx/(1+x2)2=∫0arctan(1) d(tan (y))*cos (y)4= ∫0arctan(1) dy/(cos(y))2*cos(y)4=∫0arctan(1) dy * cos(y)2=
=½∫0arctan(1) dy (1+cos(2y))=[y/2+¼sin(2y)] |0arctan(1)=
arctan(1)/2+¼sin(2*arctan(1))=arctan(1)/2+¼;
Here I used the fact that sin(2 arctan(y))=2y/(1+y2); arctan(1)=pi/4, hence
 
Answer: pi/8+1/4