Calculus Examples

$\int \frac{4 t^{3} - t^{2} + 16 t}{t^{2} + 4} d t$

Step 1

Divide $4 t^{3} - t^{2} + 16 t$ by $t^{2} + 4$ .

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Step 1.1

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

$t^{2}$

$0 t$

$4$

$4 t^{3}$

$t^{2}$

$16 t$

$0$

Step 1.2

Divide the highest order term in the dividend $4 t^{3}$ by the highest order term in divisor $t^{2}$ .

$4 t$

$t^{2}$

$0 t$

$4$

$4 t^{3}$

$t^{2}$

$16 t$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$4 t$

$t^{2}$

$0 t$

$4$

$4 t^{3}$

$t^{2}$

$16 t$

$0$

$4 t^{3}$

$0$

$16 t$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 t^{3} + 0 + 16 t$

$4 t$

$t^{2}$

$0 t$

$4$

$4 t^{3}$

$t^{2}$

$16 t$

$0$

$4 t^{3}$

$0$

$16 t$

Step 1.5

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

$4 t$

$t^{2}$

$0 t$

$4$

$4 t^{3}$

$t^{2}$

$16 t$

$0$

$4 t^{3}$

$0$

$16 t$

$t^{2}$

$0$

Step 1.6

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

						$4 t$
$t^{2}$	+	$0 t$	+	$4$		$4 t^{3}$	-	$t^{2}$	+	$16 t$	+	$0$
					-	$4 t^{3}$	-	$0$	-	$16 t$
							-	$t^{2}$	+	$0$	+	$0$

Step 1.7

Divide the highest order term in the dividend $- t^{2}$ by the highest order term in divisor $t^{2}$ .

						$4 t$	-	$1$
$t^{2}$	+	$0 t$	+	$4$		$4 t^{3}$	-	$t^{2}$	+	$16 t$	+	$0$
					-	$4 t^{3}$	-	$0$	-	$16 t$
							-	$t^{2}$	+	$0$	+	$0$

Step 1.8

Multiply the new quotient term by the divisor.

						$4 t$	-	$1$
$t^{2}$	+	$0 t$	+	$4$		$4 t^{3}$	-	$t^{2}$	+	$16 t$	+	$0$
					-	$4 t^{3}$	-	$0$	-	$16 t$
							-	$t^{2}$	+	$0$	+	$0$
							-	$t^{2}$	+	$0$	-	$4$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- t^{2} + 0 - 4$

						$4 t$	-	$1$
$t^{2}$	+	$0 t$	+	$4$		$4 t^{3}$	-	$t^{2}$	+	$16 t$	+	$0$
					-	$4 t^{3}$	-	$0$	-	$16 t$
							-	$t^{2}$	+	$0$	+	$0$
							+	$t^{2}$	-	$0$	+	$4$

Step 1.10

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

						$4 t$	-	$1$
$t^{2}$	+	$0 t$	+	$4$		$4 t^{3}$	-	$t^{2}$	+	$16 t$	+	$0$
					-	$4 t^{3}$	-	$0$	-	$16 t$
							-	$t^{2}$	+	$0$	+	$0$
							+	$t^{2}$	-	$0$	+	$4$
											+	$4$

Step 1.11

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Reorder $t^{2}$ and $4$ .

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

Step 8.2

Rewrite $4$ as $2^{2}$ .

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

Step 9

The integral of $\frac{1}{2^{2} + t^{2}}$ with respect to $t$ is $\frac{1}{2} \arctan (\frac{t}{2}) + C$ .

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

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $\arctan (\frac{t}{2})$ .

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

Step 10.2

Simplify.

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

Step 10.3

Simplify.

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Step 10.3.1

Combine $4$ and $\frac{\arctan (\frac{t}{2})}{2}$ .

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

Step 10.3.2

Cancel the common factor of $4$ and $2$ .

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Step 10.3.2.1

Factor $2$ out of $4 \arctan (\frac{t}{2})$ .

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

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $2$ out of $2$ .

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

Step 10.3.2.2.2

Cancel the common factor.

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

Step 10.3.2.2.3

Rewrite the expression.

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

Step 10.3.2.2.4

Divide $2 \arctan (\frac{t}{2})$ by $1$ .

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

Step 10.4

Reorder terms.

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