Question

∫ x tan − 1 x from 0 to 1

Answer 1

Let I=∫xtan

−1

xdx

Taking tan

−1

x as first function and x as second function and integrating by parts, we obtain

I=tan

−1

x∫xdx−∫{(

dx

d

tan

−1

x)∫xdx}dx

=tan

−1

x(

2

x

2

)−∫

1+x

2

1

⋅

2

x

2

dx

=

2

x

2

tan

−1

x

−

2

1

∫

1+x

2

x

2

dx

=

2

x

2

tan

−1

x

−

2

1

∫(

1+x

2

x

2

+1

−

1+x

2

1

)dx

=

2

x

2

tan

−1

x

−

2

1

∫(1−

1+x

2

1

)dx

=

2

x

2

tan

−1

x

−

2

1

(x−tan

−1

x)+C

=

2

x

tan

−1

x−

2

x

+

2

1

tan

−1

x+C

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