Questions: Evaluate the integral. [ int t^5left(7+t^6right)^2 d t int t^5left(7+t^6right)^2 d t= ]

Solution Steps

To evaluate the integral \(\int t^{5}(7+t^{6})^{2} \, dt\), we can use substitution. Let \( u = 7 + t^6 \), then \( du = 6t^5 \, dt \). This substitution simplifies the integral into a form that is easier to integrate.

We start with the integral \[ \int t^{5}(7+t^{6})^{2} \, dt. \] We use the substitution \( u = 7 + t^6 \), which gives us \( du = 6t^5 \, dt \) or \( dt = \frac{du}{6t^5} \). This transforms our integral into a simpler form.

Substituting \( u \) into the integral, we have: \[ \int t^{5}(u)^{2} \cdot \frac{du}{6t^5} = \frac{1}{6} \int u^{2} \, du. \] Now, we can integrate \( u^{2} \): \[ \frac{1}{6} \cdot \frac{u^{3}}{3} = \frac{u^{3}}{18}. \]

Substituting back \( u = 7 + t^6 \), we get: \[ \frac{(7 + t^6)^{3}}{18}. \] Expanding this expression, we find: \[ \frac{1}{18} \left( 343 + 147t^6 + 21t^{12} + t^{18} \right). \] Thus, the integral evaluates to: \[ \frac{t^{18}}{18} + \frac{49t^{12}}{6} + \frac{49t^{6}}{6} + C, \] where \( C \) is the constant of integration.

Final Answer