Calculus Examples

$\frac{d s}{d t} = 36 t {(9 t^{2} - 7)}^{3}$ , $s (1) = 3$

Step 1

Rewrite the equation.

$d s = 36 t {(9 t^{2} - 7)}^{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 36 t {(9 t^{2} - 7)}^{3} d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 36 t {(9 t^{2} - 7)}^{3} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $36$ is constant with respect to $t$ , move $36$ out of the integral.

$s + C_{1} = 36 \int t {(9 t^{2} - 7)}^{3} d t$

Step 2.3.2

Let $u = 9 t^{2} - 7$ . Then $d u = 18 t d t$ , so $\frac{1}{18} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 9 t^{2} - 7$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $9 t^{2} - 7$ .

$\frac{d}{d t} [9 t^{2} - 7]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $9 t^{2} - 7$ with respect to $t$ is $\frac{d}{d t} [9 t^{2}] + \frac{d}{d t} [- 7]$ .

$\frac{d}{d t} [9 t^{2}] + \frac{d}{d t} [- 7]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d t} [9 t^{2}]$ .

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Step 2.3.2.1.3.1

Since $9$ is constant with respect to $t$ , the derivative of $9 t^{2}$ with respect to $t$ is $9 \frac{d}{d t} [t^{2}]$ .

$9 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 7]$

Step 2.3.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$9 (2 t) + \frac{d}{d t} [- 7]$

Step 2.3.2.1.3.3

Multiply $2$ by $9$ .

$18 t + \frac{d}{d t} [- 7]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7$ with respect to $t$ is $0$ .

$18 t + 0$

Step 2.3.2.1.4.2

Add $18 t$ and $0$ .

$18 t$

Step 2.3.2.2

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

$s + C_{1} = 36 \int u^{3} \frac{1}{18} d u$

Step 2.3.3

Combine $u^{3}$ and $\frac{1}{18}$ .

$s + C_{1} = 36 \int \frac{u^{3}}{18} d u$

Step 2.3.4

Since $\frac{1}{18}$ is constant with respect to $u$ , move $\frac{1}{18}$ out of the integral.

$s + C_{1} = 36 (\frac{1}{18} \int u^{3} d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{18}$ and $36$ .

$s + C_{1} = \frac{36}{18} \int u^{3} d u$

Step 2.3.5.2

Cancel the common factor of $36$ and $18$ .

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Step 2.3.5.2.1

Factor $18$ out of $36$ .

$s + C_{1} = \frac{18 \cdot 2}{18} \int u^{3} d u$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $18$ out of $18$ .

$s + C_{1} = \frac{18 \cdot 2}{18 (1)} \int u^{3} d u$

Step 2.3.5.2.2.2

Cancel the common factor.

$s + C_{1} = \frac{18 \cdot 2}{18 \cdot 1} \int u^{3} d u$

Step 2.3.5.2.2.3

Rewrite the expression.

$s + C_{1} = \frac{2}{1} \int u^{3} d u$

Step 2.3.5.2.2.4

Divide $2$ by $1$ .

$s + C_{1} = 2 \int u^{3} d u$

Step 2.3.6

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$s + C_{1} = 2 (\frac{1}{4} u^{4} + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite $2 (\frac{1}{4} u^{4} + C_{2})$ as $2 (\frac{1}{4}) u^{4} + C_{2}$ .

$s + C_{1} = 2 (\frac{1}{4}) u^{4} + C_{2}$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Combine $2$ and $\frac{1}{4}$ .

$s + C_{1} = \frac{2}{4} u^{4} + C_{2}$

Step 2.3.7.2.2

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

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Step 2.3.7.2.2.1

Factor $2$ out of $2$ .

$s + C_{1} = \frac{2 (1)}{4} u^{4} + C_{2}$

Step 2.3.7.2.2.2

Cancel the common factors.

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Step 2.3.7.2.2.2.1

Factor $2$ out of $4$ .

$s + C_{1} = \frac{2 \cdot 1}{2 \cdot 2} u^{4} + C_{2}$

Step 2.3.7.2.2.2.2

Cancel the common factor.

$s + C_{1} = \frac{2 \cdot 1}{2 \cdot 2} u^{4} + C_{2}$

Step 2.3.7.2.2.2.3

Rewrite the expression.

$s + C_{1} = \frac{1}{2} u^{4} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $9 t^{2} - 7$ .

$s + C_{1} = \frac{1}{2} {(9 t^{2} - 7)}^{4} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$s = \frac{1}{2} {(9 t^{2} - 7)}^{4} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $3$ for $s$ in $s = \frac{1}{2} {(9 t^{2} - 7)}^{4} + K$ .

$3 = \frac{1}{2} {(9 \cdot 1^{2} - 7)}^{4} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{2} \cdot {(9 \cdot 1^{2} - 7)}^{4} + K = 3$ .

$\frac{1}{2} \cdot {(9 \cdot 1^{2} - 7)}^{4} + K = 3$

Step 4.2

Simplify each term.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$\frac{1}{2} \cdot {(9 \cdot 1 - 7)}^{4} + K = 3$

Step 4.2.1.2

Multiply $9$ by $1$ .

$\frac{1}{2} \cdot {(9 - 7)}^{4} + K = 3$

Step 4.2.2

Subtract $7$ from $9$ .

$\frac{1}{2} \cdot 2^{4} + K = 3$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Factor $2$ out of $2^{4}$ .

$\frac{1}{2} \cdot (2 \cdot 2^{3}) + K = 3$

Step 4.2.3.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2^{3}) + K = 3$

Step 4.2.3.3

Rewrite the expression.

$2^{3} + K = 3$

Step 4.2.4

Raise $2$ to the power of $3$ .

$8 + K = 3$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $8$ from both sides of the equation.

$K = 3 - 8$

Step 4.3.2

Subtract $8$ from $3$ .

$K = - 5$

Step 5

Substitute $- 5$ for $K$ in $s = \frac{1}{2} {(9 t^{2} - 7)}^{4} + K$ and simplify.

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Step 5.1

Substitute $- 5$ for $K$ .

$s = \frac{1}{2} {(9 t^{2} - 7)}^{4} - 5$

Step 5.2

Simplify each term.

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Step 5.2.1

Use the Binomial Theorem.

$s = \frac{1}{2} ({(9 t^{2})}^{4} + 4 {(9 t^{2})}^{3} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Apply the product rule to $9 t^{2}$ .

$s = \frac{1}{2} (9^{4} {(t^{2})}^{4} + 4 {(9 t^{2})}^{3} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.2

Raise $9$ to the power of $4$ .

$s = \frac{1}{2} (6561 {(t^{2})}^{4} + 4 {(9 t^{2})}^{3} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.3

Multiply the exponents in ${(t^{2})}^{4}$ .

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Step 5.2.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (6561 t^{2 \cdot 4} + 4 {(9 t^{2})}^{3} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.3.2

Multiply $2$ by $4$ .

$s = \frac{1}{2} (6561 t^{8} + 4 {(9 t^{2})}^{3} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.4

Apply the product rule to $9 t^{2}$ .

$s = \frac{1}{2} (6561 t^{8} + 4 (9^{3} {(t^{2})}^{3}) \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.5

Raise $9$ to the power of $3$ .

$s = \frac{1}{2} (6561 t^{8} + 4 (729 {(t^{2})}^{3}) \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.6

Multiply the exponents in ${(t^{2})}^{3}$ .

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Step 5.2.2.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (6561 t^{8} + 4 (729 t^{2 \cdot 3}) \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.6.2

Multiply $2$ by $3$ .

$s = \frac{1}{2} (6561 t^{8} + 4 (729 t^{6}) \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.7

Multiply $729$ by $4$ .

$s = \frac{1}{2} (6561 t^{8} + 2916 t^{6} \cdot - 7 + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.8

Multiply $- 7$ by $2916$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 6 {(9 t^{2})}^{2} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.9

Apply the product rule to $9 t^{2}$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 6 (9^{2} {(t^{2})}^{2}) {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.10

Raise $9$ to the power of $2$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 6 (81 {(t^{2})}^{2}) {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.11

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 5.2.2.11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 6 (81 t^{2 \cdot 2}) {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.11.2

Multiply $2$ by $2$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 6 (81 t^{4}) {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.12

Multiply $81$ by $6$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 486 t^{4} {(- 7)}^{2} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.13

Raise $- 7$ to the power of $2$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 486 t^{4} \cdot 49 + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.14

Multiply $49$ by $486$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 23814 t^{4} + 4 (9 t^{2}) {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.15

Multiply $9$ by $4$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 23814 t^{4} + 36 t^{2} {(- 7)}^{3} + {(- 7)}^{4}) - 5$

Step 5.2.2.16

Raise $- 7$ to the power of $3$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 23814 t^{4} + 36 t^{2} \cdot - 343 + {(- 7)}^{4}) - 5$

Step 5.2.2.17

Multiply $- 343$ by $36$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 23814 t^{4} - 12348 t^{2} + {(- 7)}^{4}) - 5$

Step 5.2.2.18

Raise $- 7$ to the power of $4$ .

$s = \frac{1}{2} (6561 t^{8} - 20412 t^{6} + 23814 t^{4} - 12348 t^{2} + 2401) - 5$

Step 5.2.3

Apply the distributive property.

$s = \frac{1}{2} (6561 t^{8}) + \frac{1}{2} (- 20412 t^{6}) + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Multiply $\frac{1}{2} (6561 t^{8})$ .

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Step 5.2.4.1.1

Combine $6561$ and $\frac{1}{2}$ .

$s = \frac{6561}{2} t^{8} + \frac{1}{2} (- 20412 t^{6}) + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.1.2

Combine $\frac{6561}{2}$ and $t^{8}$ .

$s = \frac{6561 t^{8}}{2} + \frac{1}{2} (- 20412 t^{6}) + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.2

Cancel the common factor of $2$ .

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Step 5.2.4.2.1

Factor $2$ out of $- 20412 t^{6}$ .

$s = \frac{6561 t^{8}}{2} + \frac{1}{2} (2 (- 10206 t^{6})) + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.2.2

Cancel the common factor.

$s = \frac{6561 t^{8}}{2} + \frac{1}{2} (2 (- 10206 t^{6})) + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.2.3

Rewrite the expression.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + \frac{1}{2} (23814 t^{4}) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.3

Cancel the common factor of $2$ .

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Step 5.2.4.3.1

Factor $2$ out of $23814 t^{4}$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + \frac{1}{2} (2 (11907 t^{4})) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.3.2

Cancel the common factor.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + \frac{1}{2} (2 (11907 t^{4})) + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.3.3

Rewrite the expression.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} + \frac{1}{2} (- 12348 t^{2}) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.1

Factor $2$ out of $- 12348 t^{2}$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} + \frac{1}{2} (2 (- 6174 t^{2})) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.4.2

Cancel the common factor.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} + \frac{1}{2} (2 (- 6174 t^{2})) + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.4.3

Rewrite the expression.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{1}{2} \cdot 2401 - 5$

Step 5.2.4.5

Combine $\frac{1}{2}$ and $2401$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2401}{2} - 5$

Step 5.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2401}{2} - 5 \cdot \frac{2}{2}$

Step 5.4

Combine $- 5$ and $\frac{2}{2}$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2401}{2} + \frac{- 5 \cdot 2}{2}$

Step 5.5

Combine the numerators over the common denominator.

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2401 - 5 \cdot 2}{2}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 5$ by $2$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2401 - 10}{2}$

Step 5.6.2

Subtract $10$ from $2401$ .

$s = \frac{6561 t^{8}}{2} - 10206 t^{6} + 11907 t^{4} - 6174 t^{2} + \frac{2391}{2}$