Question

Find each indefinite integral. Check some by calculator. $$\int 8 x^{4} d x$$

Short answer

\( \frac{8}{5} x^{5} + C \)

Step-by-step solution

Identify the Integral

The given integral is \[ \int 8 x^{4} dx \.\] This integral represents the antiderivative of the function f(x) = 8x^4 with respect to x.

Use the Power Rule for Integration

The Power Rule states that \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \.\] Applying this rule to our integral, we get \[ \int 8 x^{4} dx = 8 \int x^{4} dx \] \[ \text{Let}'s \text{ increase } n \text{ by } 1 \text{ to get } n+1 = 4 + 1 = 5. \.\] \[ \text{Now, divide by } n+1, \text{ which gives us } \frac{8}{5} x^5. \]

Add the Constant of Integration

After finding the antiderivative, we add the constant of integration C to the result: \[ \frac{8}{5} x^5 + C \.\] This completes the integration process.

Use a Calculator to Confirm the Result (if necessary)

To verify the result, you can use a graphing calculator or an online integral calculator. By entering the original function and the result, the calculator should display a graph that confirms the correctness of the integral.

Key concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is essentially the reverse of a derivative. It's a fundamental concept in calculus that helps in finding the original function given its rate of change. For instance, if you are given the function's derivative, such as the gradient of a slope, the antiderivative will tell you the height of the slope at any point.

During the solution of \[\int0 x^4 dx\], we aimed to discover the original function that, once derived, would give us \(8x^4\). The process of finding this original function is known as integration, a core operation in calculus, alongside differentiation. Understanding antiderivatives is key because it allows us to solve problems involving area, velocity, and other quantities where we have the rate of change but seek the original quantity.

Power Rule for Integration

The power rule for integration is a shortcut that enables us to integrate any function of the form \(x^n\) quickly and effectively. The rule is straightforward: to integrate \(x^n\), you add 1 to the exponent (n), and then divide the term by this new exponent. In the formula form, this is expressed as \[\int x^n dx = \frac{x^{n+1}}{n+1} + C,\] where C represents a constant of integration.

Let's apply this rule to our exercise. Starting with the integral \(\int 8x^4 dx\), we increased the exponent from 4 to 5 and then divided by this new exponent, resulting in a factor of \(\frac{8}{5}\) before the \(x^5\). This manipulation simplifies the integration process, allowing us to quickly find the antiderivative without complex calculations or guesswork.

Constant of Integration

The constant of integration, represented by the symbol C, plays a pivotal role in the world of indefinite integrals. When we integrate a function, we are essentially searching for all the possible antiderivatives. The constant of integration is a reflection of the fact that there is not just one but an infinite number of antiderivatives for a given function.

Consider the function \(F(x) = \frac{8}{5}x^5 + C\). The term C could be any real number, and regardless of which number we choose, differentiating \(F(x)\) will still give us back our original function \(8x^4\). This is because when a constant is differentiated, it results in zero; thus, it gets 'lost' in the differentiation process. It's essential to include the constant of integration when solving indefinite integrals to account for all possible original functions that could have led to the given rate of change.