Calculus Examples

$- \int t \ln (t) d t$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (t)$ and $d v = t$ .

$- (\ln (t) (\frac{1}{2} t^{2}) - \int \frac{1}{2} t^{2} \frac{1}{t} d t)$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{2}$ and $t^{2}$ .

$- (\ln (t) \frac{t^{2}}{2} - \int \frac{1}{2} t^{2} \frac{1}{t} d t)$

Step 2.2

Combine $\ln (t)$ and $\frac{t^{2}}{2}$ .

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{1}{2} t^{2} \frac{1}{t} d t)$

Step 2.3

Combine $\frac{1}{2}$ and $t^{2}$ .

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t^{2}}{2} \cdot \frac{1}{t} d t)$

Step 2.4

Multiply $\frac{t^{2}}{2}$ by $\frac{1}{t}$ .

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t^{2}}{2 t} d t)$

Step 2.5

Cancel the common factor of $t^{2}$ and $t$ .

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Step 2.5.1

Factor $t$ out of $t^{2}$ .

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t \cdot t}{2 t} d t)$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $t$ out of $2 t$ .

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t \cdot t}{t \cdot 2} d t)$

Step 2.5.2.2

Cancel the common factor.

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t \cdot t}{t \cdot 2} d t)$

Step 2.5.2.3

Rewrite the expression.

$- (\frac{\ln (t) t^{2}}{2} - \int \frac{t}{2} d t)$

Step 3

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{\ln (t) t^{2}}{2} - (\frac{1}{2} \int t d t))$

Step 4

By the Power Rule, the integral of $t$ with respect to $t$ is $\frac{1}{2} t^{2}$ .

$- (\frac{\ln (t) t^{2}}{2} - \frac{1}{2} (\frac{1}{2} t^{2} + C))$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2}$ and $t^{2}$ .

$- (\frac{\ln (t) t^{2}}{2} - \frac{1}{2} (\frac{t^{2}}{2} + C))$

Step 5.2

Rewrite $- (\frac{\ln (t) t^{2}}{2} - \frac{1}{2} (\frac{t^{2}}{2} + C))$ as $- (\frac{\ln (t) t^{2}}{2} - \frac{t^{2}}{4}) + C$ .

$- (\frac{\ln (t) t^{2}}{2} - \frac{t^{2}}{4}) + C$

Step 5.3

Reorder terms.

$- (\frac{1}{2} \ln (t) t^{2} - \frac{1}{4} t^{2}) + C$