Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (t)$ and $d v = 1$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $t$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $\ln (t^{2})$ by $1$ .

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

Step 3

Rewrite $\ln (t^{2})$ as $2 \ln (t)$ .

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

Step 4

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (t)$ and $d v = 1$ .

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $t$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 8

Apply the constant rule.

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

Step 9

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

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