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Answer:
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, dx = \frac{1}{5}( \arctan(x^5)) + c[/tex]
Step-by-step explanation:
Given
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, dx[/tex]
Required
Integrate
We have:
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, dx[/tex]
Let
[tex]u = x^5[/tex]
Differentiate
[tex]\frac{du}{dx} = 5x^4[/tex]
Make dx the subject
[tex]dx = \frac{du}{5x^4}[/tex]
So, we have:
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, dx[/tex]
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, \frac{du}{5x^4}[/tex]
[tex]\frac{1}{5} \int\ {\frac{1}{1 + x^{10}}} \, du[/tex]
Express x^(10) as x^(5*2)
[tex]\frac{1}{5} \int\ {\frac{1}{1 + x^{5*2}}} \, du[/tex]
Rewrite as:
[tex]\frac{1}{5} \int\ {\frac{1}{1 + x^{5)^2}}} \, du[/tex]
Recall that: [tex]u = x^5[/tex]
[tex]\frac{1}{5} \int\ {\frac{1}{1 + u^2}}} \, du[/tex]
Integrate
[tex]\frac{1}{5} * \arctan(u) + c[/tex]
Substitute: [tex]u = x^5[/tex]
[tex]\frac{1}{5} * \arctan(x^5) + c[/tex]
Hence:
[tex]\int\ {\frac{x^4}{1 + x^{10}}} \, dx = \frac{1}{5}( \arctan(x^5)) + c[/tex]