\( I = \int \sqrt { x ^ { 3 } + x ^ { 4 } } d x \)

To tackle the integral \( I = \int \sqrt{x^3 + x^4} \, dx \), you can factor out the common term under the square root for simplification. This gives you \( I = \int \sqrt{x^3(1 + x)} \, dx \), which can be rewritten as \( I = \int x^{3/2} \sqrt{1 + x} \, dx \). This form allows you to apply integration techniques like substitution or integration by parts more effectively. Moreover, to evaluate this integral accurately, you can use substitution methods. A common approach would be to let \( u = 1 + x \), simplifying your integral further. The relationship \( du = dx \) transforms the limits accordingly, making it easier to handle. Just keep an eye out for the boundaries when performing such substitutions!