Advanced book on Mathematics Olympiad

150 3 Real Analysis

3.2.8 Definite Integrals.........................................

Next, definite integrals. Here the limits of integration also play a role.

Example.Letf:[ 0 , 1 ]→Rbe a continuous function. Prove that

∫π

0

xf (sinx)dx=π

∫ π 2

0

f(sinx)dx.

Solution.We have

∫π

0

xf (sinx)dx=

∫ π 2

0

xf (sinx)dx+

∫π

π 2

xf (sinx)dx.

We would like to transform both integrals on the right into the same integral, and for that

we need a substitution in the second integral that changes the limits of integration. This

substitution should leavef(sinx)invariant, so it is natural to tryt=π−x. The integral

becomes

∫ π 2

0

(π−t)f(sint)dt.

Adding the two, we obtainπ

∫π 2

0 f(sinx)dx, as desired. 

453.Compute the integral

∫ 1

− 1

√ (^3) x
√ (^31) −x+√ (^31) +xdx.
454.Compute
∫π
0
xsinx
1 +sin^2 x
dx.
455.Letaandbbe positive real numbers. Compute
∫b
a
e
xa
−e
bx
x
dx.
456.Compute the integral

I=

∫ 1

0

√ (^32) x (^3) − 3 x (^2) −x+ 1 dx.