91Ó°ÊÓ

In calculus, finding the antiderivative is essentially the reverse of differentiation. When you find the antiderivative of a function, you are searching for a function that gives your original function when differentiated. This process is crucial when solving definite integrals.
In the problem, the aim was to find the antiderivative of the function \((2x + 4)(x^2 + 4x - 8)^3\). To make the process manageable, substitution is often used. The chosen substitution was \(u = x^2 + 4x - 8\) which simplifies the expression considerably.
With this substitution, the differential \(du\), which is \((2x + 4)dx\), aligns perfectly with part of the integrand. This makes it possible to directly integrate the function \(u^3\) with respect to \(u\).
After integrating \(u^3\), the result is \(\frac{1}{4}u^4 + C\). The last step is to substitute back \(x^2 + 4x - 8\) for \(u\), yielding the antiderivative \(\frac{1}{4}(x^2 + 4x - 8)^4 + C\). This function is known as the indefinite integral of the given expression.

Integration can sometimes be challenging, but thankfully, various techniques are available to handle complex integrations. One effective strategy is substitution, which was used in this exercise to transform a complex integrand into a simpler one.
Substitution works by replacing a part of the integrand with a single variable, such as \(u\), to simplify the integral into a form that is straightforward to integrate. This technique requires finding the derivative of \(u\) and replacing \(du\) in the integral.
In the exercise, the substitution \(u = x^2 + 4x - 8\) turned the problem into a familiar polynomial form \(u^3\), which is simple to integrate. Advanced problems might require other techniques like integration by parts or partial fraction decomposition but substitution remains a versatile and widely used method.
With practice, identifying which technique to utilize becomes more intuitive, making integration tasks much easier.

The Fundamental Theorem of Calculus is a cornerstone in the field, as it elegantly links the concepts of differentiation and integration. This theorem provides a way to compute the definite integral of a function and represents a key step in solving the given problem.
The theorem states that if \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then the definite integral of \(f\) from \(a\) to \(b\) is given by \(F(b) - F(a)\).
For the exercise, the antiderivative \(\frac{1}{4}(x^2 + 4x - 8)^4 + C\) was evaluated at the upper limit (\(x = 2\)) and the lower limit (\(x = 1\)). These values were used to calculate the definite integral.
By substituting \(2\) and \(1\) into the antiderivative, the difference \(F(b) - F(a)\) yielded the result \(\frac{175}{4}\), which is the value of the definite integral. This theorem not only provides an efficient way to calculate definite integrals but also highlights the interconnectedness of calculus concepts.