Calculus Examples

$\int s \cdot 6^{s} d s$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = s$ and $d v = 6^{s}$ .

$s (\frac{1}{\ln (6)} \cdot 6^{s}) - \int \frac{1}{\ln (6)} \cdot 6^{s} d s$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{\ln (6)}$ and $6^{s}$ .

$s \frac{6^{s}}{\ln (6)} - \int \frac{1}{\ln (6)} \cdot 6^{s} d s$

Step 2.2

Combine $s$ and $\frac{6^{s}}{\ln (6)}$ .

$\frac{s \cdot 6^{s}}{\ln (6)} - \int \frac{1}{\ln (6)} \cdot 6^{s} d s$

Step 3

Since $\frac{1}{\ln (6)}$ is constant with respect to $s$ , move $\frac{1}{\ln (6)}$ out of the integral.

$\frac{s \cdot 6^{s}}{\ln (6)} - (\frac{1}{\ln (6)} \int 6^{s} d s)$

Step 4

The integral of $6^{s}$ with respect to $s$ is $\frac{6^{s}}{\ln (6)}$ .

$\frac{s \cdot 6^{s}}{\ln (6)} - \frac{1}{\ln (6)} (\frac{6^{s}}{\ln (6)} + C)$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $\frac{s \cdot 6^{s}}{\ln (6)} - \frac{1}{\ln (6)} (\frac{6^{s}}{\ln (6)} + C)$ as $\frac{s \cdot 6^{s}}{\ln (6)} - \frac{6^{s}}{\ln^{2} (6)} + C$ .

$\frac{s \cdot 6^{s}}{\ln (6)} - \frac{6^{s}}{\ln^{2} (6)} + C$

Step 5.2

Reorder terms.

$\frac{1}{\ln (6)} s \cdot 6^{s} - \frac{1}{\ln^{2} (6)} \cdot 6^{s} + C$