What is the antiderivative of (16 - x)^(3/2)?

1 Answer
Jan 29, 2016

-2/5(16-x)^(3/2) + C

Explanation:

Antiderivative is just the integration. For handling this type of problem, you should be comfortable with the power rule.

color(red)(intquad x^n dx = x^(n+1)/(n+1)+C

Our problem we have to find

intquad(16-x)^(3/2) dx

We shall use a u substitution just to make into a form which we are comfortable with.

Let 16-x = u
Differentiating with respect to x on both the sides we get.

-dx= du
dx=-du

Our integral now becomes

intquadu^(3/2)(-du)
=-intquad u^(3/2)du

Use the power rule

=-u^(3/2+1)/(3/2+1) +C
=-u^(5/2)/(5/2)+C
=-2/5u^(5/2)+C

Substituting back for u we get,

=-2/5(16-x)^(3/2) + C