Evaluate the indefinite integral of 25 x^(3/2) with respect to x.

Step-by-step solution

1. 1

   Pull the constant out. Constants ride through an integral untouched: $\int 25x^{3/2}\,dx=25\int x^{3/2}\,dx.$

2. 2

   Apply the power rule with $n=\tfrac32$. The rule $\displaystyle\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C$ works for any $n\ne-1$, fractional exponents included. Here $n+1=\frac32+1=\frac52.$

3. 3

   Divide by the new exponent. $25\int x^{3/2}\,dx=25\cdot\frac{x^{5/2}}{\tfrac52}+C.$ Dividing by $\tfrac52$ is the same as multiplying by $\tfrac25$, which is where students most often slip.

4. 4

   Multiply out the constant. $25\cdot\tfrac25=10$, so the antiderivative is $10x^{5/2}+C$.

5. 5

   Check by differentiating. $\dfrac{d}{dx}\left(10x^{5/2}\right)=10\cdot\tfrac52 x^{3/2}=25x^{3/2}$, which is the original integrand.