Find the antiderivative int (2-6z)^14dz square

To find the antiderivative of the function \( (2-6z)^{14} \), we can use the substitution method. Let's set \( u = 2 - 6z \). Then, we have:<br /><br />\[ du = -6 \, dz \]<br />\[ dz = -\frac{1}{6} \, du \]<br /><br />Now, substitute \( u \) and \( dz \) into the integral:<br /><br />\[ \int (2-6z)^{14} \, dz = \int u^{14} \left(-\frac{1}{6}\right) \, du \]<br /><br />This simplifies to:<br /><br />\[ -\frac{1}{6} \int u^{14} \, du \]<br /><br />Next, we integrate \( u^{14} \):<br /><br />\[ \int u^{14} \, du = \frac{u^{15}}{15} \]<br /><br />Putting it all together, we get:<br /><br />\[ -\frac{1}{6} \cdot \frac{u^{15}}{15} = -\frac{1}{90} u^{15} \]<br /><br />Finally, substitute back \( u = 2 - 6z \):<br /><br />\[ -\frac{1}{90} (2 - 6z)^{15} \]<br /><br />Therefore, the antiderivative\[ \boxed{-\frac{1}{90} (2 - 6z)^{15} + C} \]<br /><br />where \( C \) is the constant of integration.