Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{16} x^{-5 / 4} d x$$

Short answer

Question: Evaluate the definite integral $$\int_{1}^{16} x^{-5 / 4} d x$$ using the Fundamental Theorem of Calculus. Answer: The value of the definite integral is 4.

Step-by-step solution

Find the antiderivative of x^{-5/4}

To find the antiderivative, we use the power rule of integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C $$ Here, n = -5/4. Using the power rule, we obtain the antiderivative of x^{-5/4} as: $$\int x^{-5/4} dx = \frac{x^{-5/4 + 1}}{-5/4 + 1} + C = \frac{4x^{1/4}}{1} + C = 4x^{1/4}+C$$

Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative of the given function, we can apply the Fundamental Theorem of Calculus. To evaluate the definite integral, we subtract the value of the antiderivative at the lower bound from the value at the upper bound: $$\int_{1}^{16} x^{-5/4} dx = 4x^{1/4} \Bigg|_1^{16} = 4(16^{1/4}) - 4(1^{1/4})$$

Evaluate the definite integral

Finally, we can evaluate the values of the antiderivative at the bounds and complete the calculation: $$4(16^{1/4}) - 4(1^{1/4}) = 4(2) - 4(1) = 8 - 4 = 4$$ So, the definite integral is: $$\int_{1}^{16} x^{-5/4} dx = 4$$

Key concepts

Definite Integrals

When dealing with calculus, definite integrals are an essential concept. They allow you to determine the total accumulation of a quantity, such as area under a curve, between specific bounds.
Think of a definite integral as a method to find the total value from starting point to an ending point. In the exercise, we evaluated the definite integral from 1 to 16. This means we're calculating the total area under the curve of the function from 1 to 16.
The result is not influenced by any constants of integration because these constants would cancel each other out when evaluating both bounds. The Fundamental Theorem of Calculus is the key tool used here, as it links antiderivatives and definite integrals. The calculated antiderivative helps find this accumulated value.

Antiderivatives

An antiderivative is a function that reverses the process of differentiation. In simpler terms, it's like working backwards from a derivative to find the original function.
In the problem above, the antiderivative of the function \(x^{-5/4}\) is determined using integration rules. Specifically, we're using the power rule of integration to find it.

• The antiderivative of a function \(f(x)\) is denoted as \(F(x)\).
• This means that when you differentiate \(F(x)\), you should return to \(f(x)\).

The reason we find the antiderivative is to apply the Fundamental Theorem of Calculus, which enables calculating the definite integral. Thus, the antiderivative is not the final answer but a crucial step toward it.

Power Rule of Integration

The power rule of integration is a basic principle to determine antiderivatives of powers of \(x\). It's a straightforward and vital piece in solving many calculus problems.
For a function \(x^n\), the power rule can be expressed as:\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]where \(n\) is any real number except -1 (due to division by zero).
In this exercise, \(n\) was \(-5/4\). Using the power rule, this transformed the function into an antiderivative form, \(4x^{1/4} + C\). Here,

• Add one to \(n\), resulting in \(x^{1/4}\).
• Divide and simplify the coefficient, leading to \(4\).

This rule is often one of the first integration techniques learned and is foundational for understanding more complex calculus problems.