If $f(x)=\int_{0}^{x}\left(25-t^{2}\right) e^{t^{5}} d t$, find the largest interval on which $f$ is increasing.
Answer (in intervallnotation):

Step 1 ：Given the function \(f(x)=\int_{0}^{x}\left(25-t^{2}\right) e^{t^{5}} d t\), we need to find the largest interval on which \(f\) is increasing.

Step 2 ：The function \(f(x)\) is increasing where its derivative \(f'(x)\) is positive. The Fundamental Theorem of Calculus tells us that the derivative of \(f(x)\) is just the integrand evaluated at \(x\), i.e., \(f'(x) = (25-x^2)e^{x^5}\). We need to find where this derivative is positive.

Step 3 ：The critical points are at \(x = -5\) and \(x = 5\). We need to test the intervals \((-\infty, -5)\), \((-5, 5)\), and \((5, \infty)\) to see where \(f'(x)\) is positive.

Step 4 ：The function \(f(x)\) is increasing on the interval \((-5, 5)\).

Step 5 ：Final Answer: The largest interval on which \(f\) is increasing is \((-5, 5)\). So, the final answer is \(\boxed{(-5, 5)}\).