Question

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$

Short answer

Question: Determine the indefinite integral of the vector-valued function r(t) = 〈 5t^(-4) - t^2, t^6 - 4t^3, 2/t〉. Answer: The indefinite integral of the given vector-valued function r(t) is: ∫ r(t) dt = 〈 -5/3 t^(-3) - 1/3 t^3 + C1, 1/7 t^7 - t^4 + C2, 2 ln|t| + C3〉.

Step-by-step solution

Integrate the First Component

To compute the indefinite integral of the first component function, we will integrate: $$5t^{-4} - t^2\ dt$$ Now, applying power rule for integration, we get: $$\int(5t^{-4} - t^2) dt = 5\int t^{-4} dt - \int t^2 dt = 5\frac{t^{-3}}{-3} - \frac{t^3}{3} + C_1$$ Which simplifies to: $$-\frac{5}{3}t^{-3} - \frac{1}{3}t^3 + C_1$$

Integrate the Second Component

To compute the indefinite integral of the second component function, we will integrate: $$t^6 - 4t^3\ dt$$ Now, applying the power rule for integration, we get: $$\int(t^6 - 4t^3) dt = \int t^6 dt - 4\int t^3 dt = \frac{t^7}{7} - 4\frac{t^4}{4} + C_2$$ Which simplifies to: $$\frac{1}{7}t^7 - t^4 + C_2$$

Integrate the Third Component

To compute the indefinite integral of the third component function, we will integrate: $$\frac{2}{t}\ dt$$ Now, applying the power rule for integration, we get: $$\int\frac{2}{t} dt = 2\int t^{-1} dt = 2(\ln|t|) + C_3$$

Combine the Results

Now, we combine the indefinite integrals for each component in a single vector function to get the indefinite integral of the given vector-valued function r(t): $$\int \mathbf{r}(t) dt = \left\langle -\frac{5}{3}t^{-3} - \frac{1}{3}t^3 + C_1 , \frac{1}{7}t^7 - t^4 + C_2 , 2(\ln|t|) + C_3 \right\rangle$$

Key concepts

Vector Valued Functions

When dealing with functions that return vectors instead of simple scalar values, we refer to them as vector valued functions. These functions let us handle quantities defined in more than one dimension. In mathematical terms, a vector valued function is often written in the form \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \), where \( f(t) \), \( g(t) \), and \( h(t) \) are each component functions of \( t \).

These functions can represent a range of concepts, from physical paths in space to abstract multidimensional data. The main idea is that each component of the vector can be treated as a separate function of the same independent variable \( t \).

To find the indefinite integral of a vector valued function, we independently integrate each of its components. This means treating each function within the vector as its own separate entity during the integration process. Once each component is integrated, the results are combined back into a vector form.

Power Rule of Integration

The power rule of integration is a fundamental tool used to find the antiderivatives of polynomial expressions. It's quite straightforward: If you have a function \( t^n \), you can integrate it by using the formula \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), as long as \( n eq -1 \).

For example, when integrating \( 5t^{-4} \), you apply the power rule separately for each term:

• Add 1 to the exponent: \( -4 + 1 = -3 \)
• Divide by the new exponent: \( \frac{5}{-3} \)

If the integral has multiple terms, apply the power rule to each term individually before summing them up, just like in our original problem.

This technique is key to solving integrals quickly and accurately, especially for polynomial and rational functions where the terms are relatively simple.

Integration of Rational Functions

Rational functions are expressions that can be written as a fraction where both the numerator and the denominator are polynomials. Integrating rational functions involves a few specific strategies depending on the function's structure.

For instance, the function \( \frac{2}{t} \) can be rewritten using negative exponents: \( 2t^{-1} \). Because the power of \( t \) is \(-1\), this is a special case and leads to a logarithmic integral. Thus, the integral of \( 2t^{-1} \) becomes \( 2 \ln|t| + C \).

When confronting rational functions, always consider rewriting them to leverage the power rule, or recognize when to employ other techniques like partial fraction decomposition or trigonometric substitution for more complex expressions. Keep in mind the key role of recognizing which rules apply to simplify the process progressively.