Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 149

With the substitutionu=x^1 a+^12 the integral becomes

−

1

a

∫

1

√

u^2 +^34

du=−

1

a

ln

(

u+

√

u^2 +

3

4

)

+C

=−

1

a

ln

(

1

xa

+

1

2

+

√

1

x^2 a

+

1

xa

+ 1

)

+C. 

444.Compute the integral
∫
( 1 + 2 x^2 )ex
2
dx.

445.Compute
∫
x+sinx−cosx− 1
x+ex+sinx
dx.

446.Find
∫
(x^6 +x^3 )^3

√

x^3 + 2 dx

447.Compute the integral
∫
x^2 + 1
x^4 −x^2 + 1

dx

448.Compute
∫ √
ex− 1
ex+ 1

dx, x > 0.

449.Find the antiderivatives of the functionf:[ 0 , 2 ]→R,

f(x)=

√

x^3 + 2 − 2

√

x^3 + 1 +

√

x^3 + 10 − 6

√

x^3 + 1.

450.For a positive integern, compute the integral
∫
xn
1 +x+x
2
2 !+···+

xn

n!

dx.

451.Compute the integral
∫
dx
( 1 −x^2 )^4

√

2 x^2 − 1

.

452.Compute
∫
x^4 + 1
x^6 + 1

dx.

Give the answer in the formαarctanP(x)Q(x)+C,α∈Q, andP (x), Q(x)∈Z[x].