How to solve these two tricky integrals?

[European]

I cant' solve this two integrals :

$$\int$$ (ln x)$$^{2}$$

$$\int$$ cos$$^{4}$$(x)

 

[Ataman]

You do both of them by parts.

$$\int$$ (ln x)$$^{2}$$

You can rewrite this as:

$$\int (lnx)(lnx) dx$$

with

u = lnx
dv = lnx dx


To integrate lnx dx, you have to do it by parts again. After that, it is very simple.

$$\int$$ cos$$^{4}$$(x)

Again, you can rewrite this integral as something you could do by parts.

$$\int cos^{3}xcosx dx$$

u = cos^{3}x
dv = cosx dx

-Ataman

 

[sylar]

The answer of the first question is x*((ln x)^2) - 2x*(ln x) + 2x , you can check your answer.

As for the second one, my approach would be to write the integrand as
((cos x)^2)*(1 - ((sin x)^2)) , and then finish this off by using the trigonometric identities for (cos x)^2 and sin 2x .

Lastly, try to use as many problems as you can in your spare time and take notes for choosing the most suitable method in a problem you encounter.

 

[European]

Hi , thank you very much for the answers !

By the way , I just can't solve another one :

$$\int$$( x$$^{2}$$-2x+3)lnx dx

 

[HallsofIvy]

Again, straight forward by parts: let u= ln(x), dv= (x²- 2x+ 3)dx.