1 Answer

Rabensky · Answer 1 · 2022-05-12T04:51:41+0000

Answer:

$\int\ {(x^4)/(1 + x^(10))} \, dx = (1)/(5)( \arctan(x^5)) + c$

Explanation:

Given

$\int\ {(x^4)/(1 + x^(10))} \, dx$

Required

Integrate

We have:

$\int\ {(x^4)/(1 + x^(10))} \, dx$

Let

u = x^5

Differentiate

(du)/(dx) = 5x^4

Make dx the subject

dx = (du)/(5x^4)

So, we have:

$\int\ {(x^4)/(1 + x^(10))} \, dx$

$\int\ {(x^4)/(1 + x^(10))} \, (du)/(5x^4)$

$(1)/(5) \int\ {(1)/(1 + x^(10))} \, du$

Express x^(10) as x^(5*2)

$(1)/(5) \int\ {(1)/(1 + x^(5*2))} \, du$

Rewrite as:

$(1)/(5) \int\ {(1)/(1 + x^(5)^2))} \, du$

Recall that:
u = x^5

$(1)/(5) \int\ {(1)/(1 + u^2)}} \, du$

Integrate

$(1)/(5) * \arctan(u) + c$

Substitute:
u = x^5

$(1)/(5) * \arctan(x^5) + c$

Hence:

$\int\ {(x^4)/(1 + x^(10))} \, dx = (1)/(5)( \arctan(x^5)) + c$

Please log in or register to add a comment.