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}}duIs 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.