91Ó°ÊÓ

Understanding the concept of indefinite integrals is fundamental in calculus. An indefinite integral represents a family of functions–it is the collection of all antiderivatives of a function. When we find the integral of a function without any limits, we are essentially reversing the differentiation process.

Mathematically, if you have a function f(x), the indefinite integral is written as \[ \int f(x) \,dx \] and the result F(x) + C, where F is an antiderivative of f and C represents the constant of integration, because differentiation removes constants. In our exercise, the function being integrated is a combination of polynomials and trigonometric functions, leading us to identify the need for different integration techniques.

One of the most commonly used integration techniques is the power rule. This rule is handy when dealing with polynomial terms such as \(x^n\), where 'n' is any real number except -1.

The power rule states that, to integrate a power of x, you increase the exponent by one and then divide by this new exponent. In mathematical terms, \[\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\] provided that \(n eq -1\).

In our problem, the term \(2x^2\) can be integrated using the power rule, leading to \(\frac{2}{3}x^3\) when applying the formula above.

Integrating trigonometric functions is a crucial skill when working with periodic phenomena or wave-like functions. In particular, the sine and cosine functions are straightforward to integrate.

The integral of \(\sin x\) is \(-\cos x + C\), and similarly, the integral of \(\cos x\) is \(\sin x + C\). For our exercise, the second term \(-3\sin x\) results in \(3\cos x\) after integration, by applying the rule and taking into account the coefficient of -3.

When encountering roots, such as \(\sqrt{x}\), it is often helpful to rewrite them with fractional exponents to make integration using the power rule possible. For example, \(\sqrt{x}\) is the same as \(x^{1/2}\).

By applying the power rule for fractional exponents, \(\int x^{1/2}\,dx\), we increase the exponent by one, getting \(1/2 + 1 = 3/2\), and then divide by the new exponent to obtain \(\frac{2}{3}x^{3/2}\). In the exercise, there is a coefficient of 5 to consider, so the integrated form of \(5\sqrt{x}\) is \(\frac{10}{3}x^{3/2}\), following the same power rule for integration.