Calculus Examples

$\int \frac{4 t}{t - t^{2}} d t$

Step 1

Simplify.

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Step 1.1

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

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Step 1.1.1

Raise $t$ to the power of $1$ .

$\int \frac{4 t}{t^{1} - t^{2}} d t$

Step 1.1.2

Factor $t$ out of $t^{1}$ .

$\int \frac{4 t}{t \cdot 1 - t^{2}} d t$

Step 1.1.3

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

$\int \frac{4 t}{t \cdot 1 + t (- t)} d t$

Step 1.1.4

Factor $t$ out of $t \cdot 1 + t (- t)$ .

$\int \frac{4 t}{t (1 - t)} d t$

Step 1.2

Cancel the common factor of $t$ .

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Step 1.2.1

Cancel the common factor.

$\int \frac{4 t}{t (1 - t)} d t$

Step 1.2.2

Rewrite the expression.

$\int \frac{4}{1 - t} d t$

Step 2

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int \frac{1}{1 - t} d t$

Step 3

Let $u = 1 - t$ . Then $d u = - d t$ , so $- d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 1 - t$ . Find $\frac{d u}{d t}$ .

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Step 3.1.1

Rewrite.

$\frac{- 1}{1}$

Step 3.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$4 \int \frac{- 1}{u} d u$

Step 4

Move the negative in front of the fraction.

$4 \int - \frac{1}{u} d u$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$4 (- \int \frac{1}{u} d u)$

Step 6

Multiply $- 1$ by $4$ .

$- 4 \int \frac{1}{u} d u$

Step 7

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- 4 (\ln (| u |) + C)$

Step 8

Simplify.

$- 4 \ln (| u |) + C$

Step 9

Replace all occurrences of $u$ with $1 - t$ .

$- 4 \ln (| 1 - t |) + C$