Question

find the integral of e^2x/((1+e^x)^(1/4) ) dx

Answer 1

Answer:

4/7 (eˣ+1)⁷/⁴ -4/3 (eˣ+1)³/⁴ +C

Step-by-step explanation:

To find ---> ∫e²ˣ /(1+eˣ)¹/⁴ dx

-----------

Solution--->

-------------- Let

e²ˣ

I= ∫------------------- dx

(1 + eˣ)¹/⁴

eˣ eˣ

=∫--------------------- dx

(1+ eˣ)¹/⁴

Let eˣ+1 = t

eˣ = t - 1

Differentiating with respect to x

(eˣ-0) dx = dt

eˣ dx =dt

(t -1)

I =∫----------------- dt

t¹/⁴

t 1

=∫(-------- - --------)dt

t¹/⁴ t¹/⁴

=∫t t⁻¹/⁴ dt -∫t⁻¹/⁴ dt

=∫t³/⁴ dt - ∫ t⁻¹/⁴ dt

t³/⁴⁺¹ t⁻¹/⁴ ⁺¹

=-------------- - ----------------- +C

( 3/4)+1 -(1/4)+1

t⁷/⁴ t³/⁴

=------- - ------- +C

7/4 3/4

4 4

=----- t⁷/⁴ - -----t³/⁴ + C

7 3

4 4

=-----(eˣ + 1)⁷/⁴ - -----(eˣ + 1)³/⁴ + C

7 3