91Ó°ÊÓ

One powerful way to evaluate complex integrals is by using integration techniques. There are various methods such as substitution, partial fractions, and integration by parts.
In this exercise, we are using **integration by parts**. This method is especially useful when we have an integral of a product of two functions. The formula for integration by parts is:
\[ \int u dv = uv - \int v du \]
To use this formula, we need to identify the parts of the integral. We pick one part to differentiate and the other to integrate.
For the integral \int \frac{13t^2 - 48}{\sqrt[5]{4t + 7}} dt, we set \*u* = \frac{13t^2 - 48}{\sqrt[5]{4t + 7}} and \*dv* = *dt*.
This method helps break down more complicated integrals into simpler, more manageable pieces.

When differentiating a product of functions, we use the product rule. Here’s the product rule formula:
\[ (fg)' = f'g + fg' \]
Where \*f* and *g* are functions of \*t* and \*f'* and *g'* are their respective derivatives.
In the context of our integral, we have:
*g(t)* = 13t^2 - 48
*h(t)* = (4t + 7)^{-1/5}
To find the derivative of *u*, we use the product rule:
\*du* = (13t^2 - 48)'(4t + 7)^{-1/5} + (4t + 7)^{-1/5} (13t^2 - 48)'
This breaks the differentiation of a complex function into simpler components.
This approach clarifies the differentiation process, which is vital to then substitute for the integration by parts formula.

When working with integrals, you may encounter both indefinite and definite integrals.
In the current exercise, we are working with an indefinite integral (no specific bounds).
However, understanding definite integrals is essential as they represent the area under a curve between two points.
The notation for a definite integral is:
\[ \int_{a}^{b} f(x) dx \] where '*a*' and '*b*' are the limits of integration.
To evaluate definite integrals, we first find the indefinite integral and then apply the limits:
\|F(b) - F(a)|
Integration techniques like u-substitution, and integration by parts used here, can also be applied to definite integrals.
This ensures a wide array of tools for solving various kinds of integral problems.