Math / Calculus

Question(1 point) $\int_{1}^{4} \frac{x^{5}+3}{x} d x=$ $\square$ Preview My Answers Submit Answers

Studdy Solution

STEP 1

What is this asking? Calculate the definite integral of $(x^5 + 3)/x$ from $x = 1$ to $x = 4$ . Watch out! Don't forget to apply the limits of integration after finding the indefinite integral!

STEP 2

1. Simplify the integrand
2. Find the indefinite integral
3. Evaluate the definite integral

STEP 3

We can rewrite our integrand $\frac{x^5 + 3}{x}$ as the sum of two simpler fractions: $\frac{x^5}{x} + \frac{3}{x}$ .
This makes it easier to integrate!

STEP 4

Now, let's simplify each term. $\frac{x^5}{x}$ simplifies to $x^4$ (since $x^5$ divided by $x$ is $x^{5-1} = x^4$ ).
The second term, $\frac{3}{x}$ , stays as it is.
So, our simplified integrand is $x^4 + \frac{3}{x}$ .
Much nicer, right?

STEP 5

Let's integrate $x^4 + \frac{3}{x}$ with respect to $x$ .
Remember the power rule: the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ .
Applying this to $x^4$ , we get $\frac{x^{4+1}}{4+1} = \frac{x^5}{5}$ .
The integral of $\frac{1}{x}$ is $\ln|x|$ , so the integral of $\frac{3}{x}$ is $3 \cdot \ln|x| = 3\ln|x|$ .
Don't forget to add the constant of integration, $C$ !
So, our indefinite integral is $\frac{x^5}{5} + 3\ln|x| + C$ .

STEP 6

Now, let's evaluate the definite integral from $x = 1$ to $x = 4$ .
We'll substitute these values into our indefinite integral and subtract the result at the lower limit from the result at the upper limit.

STEP 7

Substituting $x = 4$ , we get $\frac{4^5}{5} + 3\ln|4| = \frac{1024}{5} + 3\ln 4$ .

STEP 8

Substituting $x = 1$ , we get $\frac{1^5}{5} + 3\ln|1| = \frac{1}{5} + 3 \cdot 0 = \frac{1}{5}$ (since $\ln 1 = 0$ ).

STEP 9

Finally, we subtract the result at the lower limit from the result at the upper limit: $(\frac{1024}{5} + 3\ln 4) - \frac{1}{5} = \frac{1023}{5} + 3\ln 4$ .
Since $4 = 2^2$ , we can rewrite $3\ln 4$ as $3\ln(2^2) = 3 \cdot 2\ln 2 = 6\ln 2$ .
So, our final result is $\frac{1023}{5} + 6\ln 2$ .

STEP 10

The definite integral is $\frac{1023}{5} + 6\ln 2$ .

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