U substitution

[Imum Coeli]

I have a bunch of integration questions, some of which have a given u substitution. I have found them all easy except for just one.

Question:
Integrate e^(5x)/(16+e^(10x)) using substitution where u=1/4*e^5x. Write the integrand as a function of u and hence solve the integral as a function of u. Write the integral as a function of the original variable.

Notes:
If u = 1/4*e^5x then dx = 5/4*e^5x du. I am having trouble writing the integrand as a function of u. I cannot seem to simplify it to a from that I can integrate.

Thanks.

 

[tkhunny]

How about u = 4e^(5x)?

 

[Imum Coeli]

I solved it using 1/4*e^(5x) and a trig substitution.

 

[Deleted member 4993]

I have a bunch of integration questions, some of which have a given u substitution. I have found them all easy except for just one.

Question:
Integrate e^(5x)/(16+e^(10x)) using substitution where u=1/4*e^5x. Write the integrand as a function of u and hence solve the integral as a function of u. Write the integral as a function of the original variable.

Notes:
If u = 1/4*e^5x then dx = 5/4*e^5x du. I am having trouble writing the integrand as a function of u. I cannot seem to simplify it to a from that I can integrate.

Thanks.

u = 1/4 * e^5x → e^10x = 16*u²

du = 5/4 * e^5x dx → dx = du/(5*u)

Then

e^5x/(16+e^10x) dx = 4*u/(16 + 16u²) * 1/(5*u) du = 1/20 * 1/(1 + u²) du