Calculus Examples

$\int \frac{1 + 10 t^{7}}{9 t} d t$

Step 1

Since $\frac{1}{9}$ is constant with respect to $t$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{9} \int \frac{1 + 10 t^{7}}{t} d t$

Step 2

Reorder $1$ and $10 t^{7}$ .

$\frac{1}{9} \int \frac{10 t^{7} + 1}{t} d t$

Step 3

Divide $10 t^{7} + 1$ by $t$ .

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Step 3.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

Step 3.2

Divide the highest order term in the dividend $10 t^{7}$ by the highest order term in divisor $t$ .

$10 t^{6}$

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

Step 3.3

Multiply the new quotient term by the divisor.

$10 t^{6}$

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

$10 t^{7}$

$0$

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in $10 t^{7} + 0$

$10 t^{6}$

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

$10 t^{7}$

$0$

Step 3.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$10 t^{6}$

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

$10 t^{7}$

$0$

Step 3.6

Pull the next term from the original dividend down into the current dividend.

$10 t^{6}$

$t$

$0$

$10 t^{7}$

$0 t^{6}$

$0 t^{5}$

$0 t^{4}$

$0 t^{3}$

$0 t^{2}$

$0 t$

$1$

$10 t^{7}$

$0$

$0 t^{5}$

$0 t^{4}$

Step 3.7

The final answer is the quotient plus the remainder over the divisor.

$\frac{1}{9} \int 10 t^{6} + \frac{1}{t} d t$

Step 4

Split the single integral into multiple integrals.

$\frac{1}{9} (\int 10 t^{6} d t + \int \frac{1}{t} d t)$

Step 5

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$\frac{1}{9} (10 \int t^{6} d t + \int \frac{1}{t} d t)$

Step 6

By the Power Rule, the integral of $t^{6}$ with respect to $t$ is $\frac{1}{7} t^{7}$ .

$\frac{1}{9} (10 (\frac{1}{7} t^{7} + C) + \int \frac{1}{t} d t)$

Step 7

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

$\frac{1}{9} (10 (\frac{1}{7} t^{7} + C) + \ln (| t |) + C)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{7}$ and $t^{7}$ .

$\frac{1}{9} (10 (\frac{t^{7}}{7} + C) + \ln (| t |) + C)$

Step 8.2

Simplify.

$\frac{1}{9} (\frac{10 t^{7}}{7} + \ln (| t |)) + C$

Step 8.3

Reorder terms.

$\frac{1}{9} (\frac{10}{7} t^{7} + \ln (| t |)) + C$