Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

I=int(ln x)^(2)dx

$I=\int(\ln x)^{2} d x$

View step-by-step help

View step-by-step help

Find indefinite integrals using the substitution and by parts

Full solution

Q.

I=\int(\ln x)^{2} d x

Recognize Problem: Recognize that the integral $I = \int (\ln x)^2 \, dx$ is not a standard integral and requires integration by parts.
Integration by Parts: Use integration by parts, which states that $\int u \, dv = uv - \int v \, du$ . Choose $u = (\ln x)^2$ (which we will differentiate) and $dv = dx$ (which we will integrate).
Differentiate $u$ : Differentiate $u$ to find $du$ . Since $u = (\ln x)^2$ , we use the chain rule to find $du = 2(\ln x)(1/x) dx = 2(\ln x)/x dx$ .
Integrate $dv$ : Integrate $dv$ to find $v$ . Since $dv = dx$ , we have $v = x$ .
Apply Formula: Apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$ . Substituting the chosen $u$ , $v$ , $du$ , and $dv$ , we get $I = x(\ln x)^2 - \int x(2(\ln x)/x) \, dx$ .
Simplify Integral: Simplify the integral $\int x\left(\frac{2(\ln x)}{x}\right) dx$ to $\int 2(\ln x) dx$ .
Recognize New Problem: Recognize that the integral $\int 2(\ln x) \, dx$ is another integration by parts problem. Choose $u = \ln x$ and $dv = 2 \, dx$ .
Differentiate $u$ : Differentiate $u$ to find $du$ . Since $u = \ln x$ , we have $du = (\frac{1}{x}) dx$ .
Integrate $dv$ : Integrate $dv$ to find $v$ . Since $dv = 2 dx$ , we have $v = 2x$ .
Apply Formula: Apply the integration by parts formula again: $\int u \, dv = uv - \int v \, du$ . Substituting the chosen $u$ , $v$ , $du$ , and $dv$ , we get $\int 2(\ln x) \, dx = 2x(\ln x) - \int \frac{2x}{x} \, dx$ .
Simplify Integral: Simplify the integral $\int \frac{2x}{x} \, dx$ to $\int 2 \, dx$ .
Integrate: Integrate $\int 2 \, dx$ to get $2x$ .
Combine Parts: Combine all parts to write the final expression for the original integral $I$ . We have $I = x(\ln x)^2 - (2x(\ln x) - 2x) + C$ , where $C$ is the constant of integration.
Final Expression: Simplify the expression to combine like terms. We have $I = x(\ln x)^2 - 2x(\ln x) + 2x + C$ .

More problems from Find indefinite integrals using the substitution and by parts

Question

Evaluate the integral.

\newline

\int-x 4^{-3 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{3} e^{-2 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int x 5^{3 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int-2 x e^{2 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int-2 x \cos (-2 x) d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int-x \cos (-3 x) d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 5^{3 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int-6 x 4^{-4 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 2^{3 x} d x

\newline

Posted 3 months ago

Question

Evaluate the integral.

\newline

\int-3 x \sin (-2 x) d x

\newline

Posted 3 months ago