Problem 46 - Calculus and its applications

Question

Find each integral. $$ \int\left(\frac{4}{\sqrt[5]{x}}+\frac{3}{4} e^{6 x}-\frac{7}{x}\right) d x, x>0 $$

Step-by-Step Solution

Answer

\( \frac{20}{3}x^{4/5} + \frac{1}{8} e^{6x} - 7 \ln|x| + C \)

Step 1: Identify the Integral

Evaluate:
Find each integral.
$$
\int\left(\frac{4}{\sqrt[5]{x}}+\frac{3}{4} e^{6 x}-\frac{7}{x}\right) d x,
x>0
$$

Step 2: Rewrite and Split

Split into terms and rewrite roots as fractional exponents.

Step 3: Apply Integration Rules

• \(\int x^n dx = x^{n+1}/(n+1)+C\)
• \(\int 1/x\,dx=\ln|x|+C\)
• \(\int e^{ax}dx=e^{ax}/a+C\)

Step 4: Result

\( \frac{20}{3}x^{4/5} + \frac{1}{8} e^{6x} - 7 \ln|x| + C \)

Key Concepts

Exponential Functions

Exponential functions are fundamental in calculus and appear frequently in integration problems. They are expressed in the form \( e^{k x} \), where \( e \) is the base of natural logarithms (approximately 2.718), and \( k \) is a constant. These functions grow or decay at exponential rates, depending on the sign and value of \( k \). Understanding exponential functions is key for effective integration, as they have specific rules:

• The integral of \( e^{kx} \) is \( \frac{1}{k}e^{kx} + C \), where \( C \) is the constant of integration.

In the context of the given exercise, the function \( \frac{3}{4} e^{6x} \) represents an exponential function with a constant factor of \( \frac{3}{4} \). When integrating, it follows the general rule for exponential integrals, leading us to \( \frac{3}{24} e^{6x} + C \), after considering the constant \( 6 \) multiplication.

Power Functions

Power functions are another vital element of calculus, usually expressed as \( x^n \), where \( n \) is any real number. Integration of power functions involves the power rule, which states:

• The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), provided that \( n eq -1 \).

In the given integration exercise, we encounter two power functions. The term \( \frac{4}{\sqrt[5]{x}} \) can be rewritten as \( 4 x^{-1/5} \), showing it clearly as a power function with \( n = -1/5 \). Applying the power rule, the integral becomes \( 4 \cdot \frac{x^{4/5}}{4/5} + C \), simplified to \( 5x^{4/5} + C \). The other term, \( -\frac{7}{x} \), equivalent to \( -7x^{-1} \), poses a special case since \( n = -1 \). For \( n = -1 \), the integral becomes \( -7 \ln|x| + C \). Thus, understanding power functions and their integration rules is crucial for solving such problems.

Integral of Polynomials

Integrating polynomials involves summing the integrals of individual terms, a straightforward application of the power function integration rule. A polynomial function is the sum of multiples of power functions. In our exercise, the integrand is a combination of exponential and power functions, but the approach with polynomial-style sums still applies:

• Each term of the integrand is handled individually, using their respective integration rules.
• Results from each separate integration are then combined to give the final answer.

When integrating polynomials (or expressions with multiple terms), it is essential to maintain the constant of integration \( C \) in your final expression. Our example showcases the separation into components: exponential and power, followed by integration and rearrangement of constants. The resulting integrated expression from the exercise is a sum involving different forms: emerged from careful application of rules to each term.