Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 151

457.Compute the integral
∫a

0

dx

x+

√

a^2 −x^2

(a > 0 ).

458.Compute the integral

∫ π 4

0

ln( 1 +tanx)dx.

459.Find
∫ 1

0

ln( 1 +x)

1 +x^2

dx.

460.Compute
∫∞

0

lnx

x^2 +a^2

dx,

whereais a positive constant.

461.Compute the integral

∫ π 2

0

xcosx−sinx

x^2 +sin^2 x

dx.

462.Letαbe a real number. Compute the integral

I(α)=

∫ 1

− 1

sinαdx

1 − 2 xcosα+x^2

.

463.Give an example of a functionf:( 2 ,∞)→( 0 ,∞)with the property that

∫∞

2

fp(x)dx

is finite if and only ifp∈[ 2 ,∞).

There are special types of integrals that are computed recursively. We illustrate this
with a proof of the Leibniz formula.

The Leibniz formula.

π

4

= 1 −

1

3

+

1

5

−

1

7

+···.