Math / Calculus

Question $\int x \ln x d x$

Studdy Solution

STEP 1

1. We are given the integral $\int x \ln x \, dx$ .
2. We will use integration by parts to solve this integral.
3. Integration by parts formula: $\int u \, dv = uv - \int v \, du$ .

STEP 2

1. Identify $u$ and $dv$ for integration by parts.
2. Differentiate $u$ to find $du$ and integrate $dv$ to find $v$ .
3. Apply the integration by parts formula.
4. Simplify the resulting expression.
5. Evaluate the remaining integral if necessary.

STEP 3

Identify $u$ and $dv$ :
Let $u = \ln x$ and $dv = x \, dx$ .

STEP 4

Differentiate $u$ to find $du$ :
$du = \frac{1}{x} \, dx$
Integrate $dv$ to find $v$ :
$v = \frac{x^2}{2}$

STEP 5

Apply the integration by parts formula:
$\int x \ln x \, dx = uv - \int v \, du$
Substitute $u$ , $v$ , and $du$ into the formula:
$\int x \ln x \, dx = \left(\ln x\right)\left(\frac{x^2}{2}\right) - \int \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \, dx$

STEP 6

Simplify the expression:
$\int x \ln x \, dx = \frac{x^2 \ln x}{2} - \int \frac{x}{2} \, dx$

STEP 7

Evaluate the remaining integral:
$\int \frac{x}{2} \, dx = \frac{1}{2} \int x \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4}$

STEP 8

Substitute back to find the final result:
$\int x \ln x \, dx = \frac{x^2 \ln x}{2} - \frac{x^2}{4} + C$
where $C$ is the constant of integration.
The solution to the integral is:
$\boxed{\frac{x^2 \ln x}{2} - \frac{x^2}{4} + C}$

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