Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

intsin^(5)xcos^(5)xdx

$\int \sin ^{5} x \cos ^{5} x d x$

View step-by-step help

View step-by-step help

Power property of logarithms

Full solution

Q.

\int \sin ^{5} x \cos ^{5} x d x

Identify Substitution: Identify a substitution that can simplify the integral. $\newline$ We can use the substitution $u = \sin(x)$ , which implies $du = \cos(x)dx$ .
Rewrite in terms of u: Rewrite the integral in terms of u and du. $\newline$ The integral becomes $\int u^5(1-u^2)^2\,du$ , since $\cos^2(x) = 1 - \sin^2(x)$ and we have $\cos^5(x) = \cos(x) \cdot (1 - \sin^2(x))^2$ .
Expand Integrand: Expand the integrand. $\newline$ We need to expand $(1-u^2)^2$ to integrate term by term. The expansion is $(1 - 2u^2 + u^4)$ .
Multiply by $u^5$ : Multiply the expanded terms by $u^5$ . $\newline$ Multiplying $u^5$ by each term in the expansion gives us $u^5 - 2u^7 + u^9$ .
Integrate Each Term: Integrate each term separately. $\newline$ The integral of $u^5$ is $(u^6)/6$ , the integral of $-2u^7$ is $(-2u^8)/8$ , and the integral of $u^9$ is $(u^{10})/10$ .
Combine and Simplify: Combine the integrated terms and simplify. $\newline$ Combining the terms, we get $(u^6)/6 - (u^8)/4 + (u^{10})/10$ .
Substitute back in $x$ : Substitute back in terms of $x$ . $\newline$ Since $u = \sin(x)$ , we substitute back to get $\frac{\sin^6(x)}{6} - \frac{\sin^8(x)}{4} + \frac{\sin^{10}(x)}{10} + C$ , where $C$ is the constant of integration.

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 2 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 2 months ago

Question

Use the following function rule to find

f(1)

\newline

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

\newline

f(1) =

Posted 1 month 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 2 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 2 months ago

Question

Solve. Simplify your answer.

\newline

7\log x = 7

\newline

x =

Posted 1 month 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 2 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 2 months ago