Use the Substitution Rule to evaluate the indefinite integral.?

Question

The integral of x/(1+x^4) dx.

Answer 1

Let u = x², so du = 2xdx

So, ∫ (x/1+x^4) dx =

1/2 ∫ (1/1+u²) du =

1/2 arctan(u) + C =

1/2 arctan(x²) + C

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Answer 2

The trick is that you have to recognize that this is similar to the integral of 1 / (1 + x^2) which equals arctan(x).

Recognize the general formula for a the integral:

integral( du / (a^2 + u^2) ) = 1/a * arctan(u/a)

integral ( x / (1 + x^4 ) )

We see that u = x^2, and a = 1, therefore du = 2x dx

1/2 * integral( du / (1 + u^2) )

= 1/2 * 1/1 * artcan(u/a) + c
= 1/2 * arctan(x^2) + c

Answer 3

Let u = x^2, then du = 2x dx or (1/2)du = xdx
Thus,
the integral of x/(1+x^4) dx
= (1/2) the integral of du/(1+u^2)
= (1/2) arctan(u) + C
= (1/2) arctan(x^2) + C

Answer 4

you initiate via recognizing that y^3 is the by-product (y^4)/4; So in case you place u = (2*y^4 - a million), then du = 8*y^3; take the component of 8 exterior the crucial to get (a million/8)?[a million/?u]du; The crucial is a million/8 * 2*?u = a million/4 * ?[2*y^4 - a million] + C you are able to examine that this applies via taking the by-made from a million/4 * ?[2*y^4 - a million] + C

Answer 5

Let u = x^2, du = 2x dx
Then we get 1/2*int(du/(1+ u²)= 1/2*arctan u + c =
1/2*arctan(x^2) + c.

Answer 6

Let u = x^2 thus du = 2x dx
So ∫x/(1+x^4) dx.
= ½∫2xdx./(1+x^4)
= ½∫du./(1+u^2)
= ½ arctan(u) + c
= ½ arctan(x^2) + c