Is Substitution Allowed in Integration with Negative Exponents?

[theRukus]

Homework Statement

$\int \frac{-2x}{\sqrt[4]{x+2}}dx$

The Attempt at a Solution

$=-2*\int x(x+2)^{\frac{1}{4}}dx$

Let $u=x+2$.
Then, $u-2=x$,
and $du = dx$

.. Continued from above,
$=-2*\int (u-2)u^{\frac{1}{4}}du$
$=-2*\int u^{5/4}-2u^{\frac{1}{4}}du$Is that last step allowed?

 

[Dick]

Homework Statement

$\int \frac{-2x}{\sqrt[4]{x+2}}dx$

The Attempt at a Solution

$=-2*\int x(x+2)^{1/4}dx$

Let $u=x+2$.
Then, $u-2=x$,
and $du = dx$

.. Continued from above,
$=-2*\int (u-2)u^{1/4}du$
$=-2*\int u^{5/4}-2u^{1/4}du$


Is that last step allowed?

Yes, absolutely. You are doing it exactly correctly.

 

[SteamKing]

Except the exponent (1/4) must be negative since the factor (x+2)^(1/4) was in the denominator in the OP.

 

[Dick]

Except the exponent (1/4) must be negative since the factor (x+2)^(1/4) was in the denominator in the OP.

Ack. Missed that. Sorry.