Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Integrate $\ln[(x)(1-x)]$ in the limits $0$ to $1$

View step-by-step help

View step-by-step help

Solve advanced linear inequalities

Full solution

Q. Integrate

\ln[(x)(1-x)]

in the limits

0

to

1

Rewrite Integral: Rewrite the integral to make it easier to work with. $\int_{0}^{1} \ln(x(1-x)) \, dx$
Split Logarithm: Use the property of logarithms to split the $ln$ function. $\int_{0}^{1} (\ln(x) + \ln(1-x)) \, dx$
Split Integrals: Split the integral into two separate integrals. $\int_0^1 \ln(x) \, dx + \int_0^1 \ln(1-x) \, dx$
Integration by Parts: Use integration by parts for $\int \ln(x) \, dx$ where $u=\ln(x)$ and $dv=dx$ . Let $u = \ln(x)$ , then $du = (1/x) \, dx$ . Let $dv = dx$ , then $v = x$ . Now, integrate by parts: $\int u \, dv = uv - \int v \, du$ .
Apply Integration by Parts: Apply integration by parts to the first integral. $\newline$ $x \ln(x) - \int x \cdot (\frac{1}{x}) \, dx$ from $0$ to $1$ $\newline$ $x \ln(x) - \int 1 \, dx$ from $0$ to $1$
Simplify and Integrate: Simplify and integrate. $\int_{0}^{1} x \ln(x) - x \,dx$
Integration by Parts $\ln(1-x)$ : Now, use integration by parts for $\int \ln(1-x) \, dx$ where $u=\ln(1-x)$ and $dv=dx$ .
Let $u = \ln(1-x)$ , then $du = -\left(\frac{1}{1-x}\right) dx$ .
Let $dv = dx$ , then $v = x$ .
Now, integrate by parts: $\int u \, dv = uv - \int v \, du$ .
Apply Integration by Parts: Apply integration by parts to the second integral. $\newline$ $x \ln(1-x) + \int_0^1 x \cdot \left(\frac{1}{1-x}\right) dx$ $\newline$ $x \ln(1-x) + \int_0^1 \frac{x}{1-x} dx$
Complex Integral $\ln(1-x) \, dx$ : The integral $\int \frac{x}{1-x} \, dx$ is more complex and requires partial fractions or another method to solve. $\newline$ However, we can notice that the function $\ln(1-x)$ is not defined at $x=1$ , which means the integral does not converge.

More problems from Solve advanced linear inequalities

Question

-2 x+b y=59

\newline

5 x-y=1

\newline

If the system of equations has exactly one solution

(x, y)

x=2

, what is the value of

b

Posted 2 months ago

Question

\newline

\frac{\sqrt[3]{2}}{\sqrt[3]{-128}}=

Posted 6 days ago

Question

\newline

\frac{\sqrt[4]{2}}{\sqrt[4]{162}}=

Posted 3 months ago

Question

\newline

\sqrt[5]{-16} \cdot 2^{\frac{1}{5}}=

Posted 2 months ago

Question

y

\newline

\begin{array}{l} \frac{y-1}{3 y+7}=\frac{1}{10} \\ y=\square \end{array}

Posted 3 months ago

Question

x

\newline

Your answer must be simplified.

\newline

-30<\frac{x}{-4}

Posted 2 months ago

Question

x

\newline

Your answer must be simplified.

\newline

-36+x<8

Posted 2 months ago

Question

x

\newline

Your answer must be simplified.

\newline

2 x<15

Posted 3 months ago

Question

x

\newline

Your answer must be simplified.

\newline

x-6 \leq 3

Posted 1 month ago

Question

x

\newline

Your answer must be simplified.

\newline

7>\frac{x}{4}

Posted 3 months ago