Questions: Use integration by substitution to solve the integral below. Use C for the constant of integration. [ int-2 y^2 e^-5 y^3 d y ]

Transcript text: Use integration by substitution to solve the integral below. Use $C$ for the constant of integration. \[ \int-2 y^{2} e^{-5 y^{3}} d y \]

Solution

Solution Steps

Step 1: Identify the Substitution

Let $ u = -5y^3 $. Then, we differentiate to find $ du $: \[ du = -15y^2 dy \quad \Rightarrow \quad dy = \frac{du}{-15y^2} \]

Step 2: Rewrite the Integral

Substituting $ u $ and $ dy $ into the integral, we have: \[ \int -2 y^{2} e^{-5 y^{3}} dy = \int -2 y^{2} e^{u} \left(\frac{du}{-15y^2}\right) \] This simplifies to: \[ \int \frac{2}{15} e^{u} du \]

Step 3: Integrate

Now, we integrate: \[ \int \frac{2}{15} e^{u} du = \frac{2}{15} e^{u} + C \] Substituting back $ u = -5y^3 $: \[ \frac{2}{15} e^{-5y^3} + C \]