Evaluate the integral of 5 y^(2/5) with respect to y.

Step-by-step solution

1. 1

   Pull the constant outside. A numerical factor passes straight through an integral:

   $\int 5y^{2/5}\,dy=5\int y^{2/5}\,dy$

2. 2

   Apply the power rule for integration. For $n\neq -1$:

   $\int y^n\,dy=\frac{y^{n+1}}{n+1}+C$

   Here $n=\tfrac25$, which is safely away from the excluded value $-1$, so the rule applies.

3. 3

   Add one to the exponent. Using a common denominator of $5$:

   $\frac{2}{5}+1=\frac{2}{5}+\frac{5}{5}=\frac{7}{5}$

   $\int y^{2/5}\,dy=\frac{y^{7/5}}{7/5}$

4. 4

   Turn the division by a fraction into a multiplication. Dividing by $\tfrac75$ is multiplying by $\tfrac57$:

   $5\cdot\frac{5}{7}y^{7/5}=\frac{25}{7}y^{7/5}$

   so the antiderivative is $\tfrac{25}{7}y^{7/5}+C$.

5. 5

   Check by differentiating back. The derivative must reproduce the integrand:

   $\frac{d}{dy}\left(\frac{25}{7}y^{7/5}\right)=\frac{25}{7}\cdot\frac{7}{5}y^{2/5}=5y^{2/5}\qquad\checkmark$

   Numerically at $y=2$ the derivative evaluates to $5\cdot 2^{0.4}\approx 6.5975$, matching a numerical difference quotient of the antiderivative $\checkmark$.