Advanced book on Mathematics Olympiad

148 3 Real Analysis

We begin by recalling the basic facts about indefinite integrals. Integration is the
inverse operation to differentiation. The fundamental methods for computing integrals
are the backward application of the chain rule, which takes the form
∫
f (u(x))u′(x)dx=

∫

f (u)du

and shows up in the guise of the first and second substitutions, and integration by parts
∫
udv=uv−

∫

vdu,

which comes from the product rule for derivatives. Otherwise, there is Jacobi’s partial
fraction decomposition method for computing integrals of rational functions, as well as
standard substitutions such as the trigonometric and Euler’s substitutions.
Now let us turn to our nonstandard examples.

Example.Compute

I 1 =

∫

sinx

sinx+cosx

dx and I 2 =

∫

cosx

sinx+cosx

dx.

Solution.The well-known approach is to use the substitution tanx 2 =t. But it is much
simpler to write the system

I 1 +I 2 =

∫

sinx+cosx

sinx+cosx

dx=

∫

1 dx=x+C 1 ,

−I 1 +I 2 =

∫

cosx−sinx

sinx+cosx

dx=ln(sinx+cosx)+C 2 ,

and then solve to obtain

I 1 =

1

2

x−

1

2

ln(sinx+cosx)+C 1 ′ and I 2 =

1

2

x+

1

2

ln(sinx+cosx)+C′ 2. 

We continue with a more difficult computation based on a substitution.

Example.Fora>0 compute the integral
∫
1
x

√

x^2 a+xa+ 1

dx, x > 0.

Solution.Factor anx^2 aunder the square root to transform the integral into
∫
1
xa+^1

√

1 +x^1 a+x^12 a

dx=

∫

1

√(

1

xa+

1

2

) 2

+^34

·

1

xa+^1

dx.