/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 43

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{4}(1-x)(x-4) d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral \(\int_{1}^{4}(1-x)(x-4) dx\). Answer: \(\frac{155}{6}\).

Step by step solution

01

Simplify the function

To make the integration process easier, let's first simplify the given function by expanding the brackets: \((1-x)(x-4)\). $$ (1-x)(x-4) = 1 \cdot x - 1 \cdot (-4) - x \cdot x - x \cdot (-4) = x + 4 - x^2 + 4x = 5x - x^2 + 4. $$ Now the function we need to integrate is \(5x - x^2 + 4\).
02

Apply the Fundamental Theorem of Calculus (Part 1)

Now, we need to find the antiderivative of the function \(5x - x^2 + 4\). We'll simply integrate the function term by term: $$ \int(5x - x^2 + 4) dx = \int 5x dx - \int x^2 dx + \int 4 dx. $$ For each term, we can apply the power rule for integration. Recall that the power rule for integration states that \(\int x^n dx = \frac{1}{n+1}x^{n+1} + C\): $$ \int 5x dx - \int x^2 dx + \int 4 dx = \frac{5}{2}x^2 - \frac{1}{3}x^3 + 4x + C $$
03

Apply the Fundamental Theorem of Calculus (Part 2)

Now that we have found the antiderivative, \(\frac{5}{2}x^2 - \frac{1}{3}x^3 + 4x + C\), we can evaluate the definite integral using the limits 1 and 4: $$ \int_{1}^{4}(5x - x^2 + 4) dx = \left[\frac{5}{2}x^2 - \frac{1}{3}x^3 + 4x\right]_1^4. $$ Now we just need to plug in the limits into this expression and subtract the results: $$ \left[\frac{5}{2}(4)^2 - \frac{1}{3}(4)^3 + 4(4)\right] - \left[\frac{5}{2}(1)^2 - \frac{1}{3}(1)^3 + 4(1)\right] = \left[40 - \frac{64}{3} + 16\right] - \left[\frac{5}{2} - \frac{1}{3} + 4\right]. $$
04

Simplify and calculate the result

Now we just need to simplify and calculate the result: $$ \left[40 - \frac{64}{3} + 16\right] - \left[\frac{5}{2} -\frac{1}{3} + 4\right] = \left[\frac{120}{3} - \frac{64}{3} + \frac{48}{3}\right] - \left[\frac{5}{2} -\frac{1}{3} + \frac{12}{3}\right] = \left[\frac{104}{3}\right] - \left[\frac{16}{6} + \frac{23}{6}\right] = \frac{104}{3} - \frac{39}{6} = \frac{155}{6}. $$ So the result of the integral is $$ \int_{1}^{4}(1-x)(x-4) dx = \frac{155}{6}. $$

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