Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\int 3\sqrt[5]{x^{2}}\,dx$

View step-by-step help

View step-by-step help

Power property of logarithms

Full solution

Q.

\int 3\sqrt[5]{x^{2}}\,dx

Identify Integral: Identify the integral that needs to be solved. $\newline$ We need to solve the integral of the cube root of $5$ times $x$ squared, which is written as $\int(5^{1/3} \cdot x^2) \, dx$ .
Rewrite Integral: Rewrite the integral in a more convenient form. $\newline$ We can rewrite the integral as $5^{\frac{1}{3}} \times \int x^2 \, dx$ , since $5^{\frac{1}{3}}$ is a constant and can be factored out of the integral.
Apply Power Rule: Apply the power rule for integration. $\newline$ The power rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration. $\newline$ So, $\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$ .
Combine with Constant: Combine the constant with the antiderivative. $\newline$ Now we multiply the antiderivative by the constant factor we factored out earlier, which gives us $5^{\frac{1}{3}} \cdot \frac{x^3}{3}$ + $C$ .
Write Final Answer: Write the final answer. $\newline$ The final answer is $5^{\frac{1}{3}} \cdot \frac{x^3}{3} + C$ .

More problems from Power property of logarithms

Question

Solve. Write your answer as an integer or a fraction in simplest form.

\newline

5^x=1

\newline

x=

Posted 3 months ago

Question

Steven opened a savings account and deposited

\$100.00

as principal. The account earns

9\%

interest, compounded annually. What is the balance after

5

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

Posted 3 months ago

Question

Use the following function rule to find

f(1)

\newline

f(x) = 12(7)^x + 8

\newline

f(1) =

Posted 2 months ago

Question

The town of Valley View conducted a census this year, which showed that it has a population of

1,800

people. Based on the census data, it is estimated that the population of Valley View will grow by

12\%

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the town population,

y

x

decades after this census was taken.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y =

Posted 2 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = 9^x

\newline

x =

Posted 3 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = e^x

\newline

x =

Posted 1 month ago

Question

Solve. Simplify your answer.

\newline

\log u = 1

\newline

u =

Posted 3 months ago

Question

Solve. Simplify your answer.

\newline

7\log x = 7

\newline

x =

Posted 2 months ago

Question

g(t) = 3^t

change over the interval from

t = 3

t = 4

\newline

\newline

g(t)

3

\newline

g(t)

3

\newline

g(t)

3\%

\newline

g(t)

200\%

Posted 3 months ago

Question

f(x)

6

over every unit interval in

x

f(0) = 0

\newline

Which could be a function rule for

f(x)

\newline

\newline

f(x) = 6^x

\newline

f(x) = 6x

\newline

f(x) = \frac{1}{6^x}

\newline

f(x) = \frac{x}{6}

Posted 3 months ago