Calculus Examples

$\frac{d s}{d t} = 28 {(7 t^{2} - 5)}^{3}$ , $s (1) = 5$

Step 1

Rewrite the equation.

$d s = 28 {(7 t^{2} - 5)}^{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 28 {(7 t^{2} - 5)}^{3} d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 28 {(7 t^{2} - 5)}^{3} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $28$ is constant with respect to $t$ , move $28$ out of the integral.

$s + C_{1} = 28 \int {(7 t^{2} - 5)}^{3} d t$

Step 2.3.2

Expand ${(7 t^{2} - 5)}^{3}$ .

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Step 2.3.2.1

Use the Binomial Theorem.

$s + C_{1} = 28 \int {(7 t^{2})}^{3} + 3 {(7 t^{2})}^{2} \cdot - 5 + 3 (7 t^{2}) {(- 5)}^{2} + {(- 5)}^{3} d t$

Step 2.3.2.2

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} {(7 t^{2})}^{2} + 3 {(7 t^{2})}^{2} \cdot - 5 + 3 (7 t^{2}) {(- 5)}^{2} + {(- 5)}^{3} d t$

Step 2.3.2.3

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} (7 t^{2} (7 t^{2})) + 3 {(7 t^{2})}^{2} \cdot - 5 + 3 (7 t^{2}) {(- 5)}^{2} + {(- 5)}^{3} d t$

Step 2.3.2.4

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} (7 t^{2} (7 t^{2})) + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) {(- 5)}^{2} + {(- 5)}^{3} d t$

Step 2.3.2.5

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} (7 t^{2} (7 t^{2})) + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) + {(- 5)}^{3} d t$

Step 2.3.2.6

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} (7 t^{2} (7 t^{2})) + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 {(- 5)}^{2} d t$

Step 2.3.2.7

Rewrite the exponentiation as a product.

$s + C_{1} = 28 \int 7 t^{2} (7 t^{2} (7 t^{2})) + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.8

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 t^{2} (7 \cdot 7 t^{2} t^{2}) + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.9

Move parentheses.

$s + C_{1} = 28 \int 7 t^{2} (7 \cdot 7 t^{2}) t^{2} + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.10

Move parentheses.

$s + C_{1} = 28 \int 7 t^{2} (7 \cdot 7) t^{2} t^{2} + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.11

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 (7 t^{2} (7 t^{2})) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.12

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 (7 \cdot 7 t^{2} t^{2}) \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.13

Move parentheses.

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 (7 \cdot 7 t^{2}) t^{2} \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.14

Move parentheses.

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 (7 \cdot 7) t^{2} t^{2} \cdot - 5 + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.15

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 \cdot 7 \cdot 7 t^{2} \cdot - 5 t^{2} + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.16

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 (7 t^{2}) (- 5 \cdot - 5) - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.17

Move $t^{2}$ .

$s + C_{1} = 28 \int 7 \cdot 7 \cdot 7 t^{2} t^{2} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 (- 5 \cdot - 5) d t$

Step 2.3.2.18

Multiply $7$ by $7$ .

$s + C_{1} = 28 \int 49 \cdot 7 t^{2} t^{2} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.19

Multiply $49$ by $7$ .

$s + C_{1} = 28 \int 343 t^{2} t^{2} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.20

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s + C_{1} = 28 \int 343 t^{2 + 2} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.21

Add $2$ and $2$ .

$s + C_{1} = 28 \int 343 t^{4} t^{2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s + C_{1} = 28 \int 343 t^{4 + 2} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.23

Add $4$ and $2$ .

$s + C_{1} = 28 \int 343 t^{6} + 3 \cdot 7 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.24

Multiply $3$ by $7$ .

$s + C_{1} = 28 \int 343 t^{6} + 21 \cdot 7 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.25

Multiply $21$ by $7$ .

$s + C_{1} = 28 \int 343 t^{6} + 147 \cdot - 5 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.26

Multiply $147$ by $- 5$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{2} t^{2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.27

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{2 + 2} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.28

Add $2$ and $2$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 3 \cdot 7 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.29

Multiply $3$ by $7$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 21 \cdot - 5 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.30

Multiply $21$ by $- 5$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} - 105 \cdot - 5 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.31

Multiply $- 105$ by $- 5$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 525 t^{2} - 5 \cdot - 5 \cdot - 5 d t$

Step 2.3.2.32

Multiply $- 5$ by $- 5$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 525 t^{2} + 25 \cdot - 5 d t$

Step 2.3.2.33

Multiply $25$ by $- 5$ .

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 525 t^{2} - 125 d t$

$s + C_{1} = 28 \int 343 t^{6} - 735 t^{4} + 525 t^{2} - 125 d t$

Step 2.3.3

Split the single integral into multiple integrals.

$s + C_{1} = 28 (\int 343 t^{6} d t + \int - 735 t^{4} d t + \int 525 t^{2} d t + \int - 125 d t)$

Step 2.3.4

Since $343$ is constant with respect to $t$ , move $343$ out of the integral.

$s + C_{1} = 28 (343 \int t^{6} d t + \int - 735 t^{4} d t + \int 525 t^{2} d t + \int - 125 d t)$

Step 2.3.5

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

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) + \int - 735 t^{4} d t + \int 525 t^{2} d t + \int - 125 d t)$

Step 2.3.6

Since $- 735$ is constant with respect to $t$ , move $- 735$ out of the integral.

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) - 735 \int t^{4} d t + \int 525 t^{2} d t + \int - 125 d t)$

Step 2.3.7

By the Power Rule, the integral of $t^{4}$ with respect to $t$ is $\frac{1}{5} t^{5}$ .

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) - 735 (\frac{1}{5} t^{5} + C_{3}) + \int 525 t^{2} d t + \int - 125 d t)$

Step 2.3.8

Since $525$ is constant with respect to $t$ , move $525$ out of the integral.

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) - 735 (\frac{1}{5} t^{5} + C_{3}) + 525 \int t^{2} d t + \int - 125 d t)$

Step 2.3.9

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) - 735 (\frac{1}{5} t^{5} + C_{3}) + 525 (\frac{1}{3} t^{3} + C_{4}) + \int - 125 d t)$

Step 2.3.10

Apply the constant rule.

$s + C_{1} = 28 (343 (\frac{1}{7} t^{7} + C_{2}) - 735 (\frac{1}{5} t^{5} + C_{3}) + 525 (\frac{1}{3} t^{3} + C_{4}) - 125 t + C_{5})$

Step 2.3.11

Simplify.

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Step 2.3.11.1

Simplify.

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Step 2.3.11.1.1

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

$s + C_{1} = 28 (343 (\frac{t^{7}}{7} + C_{2}) - 735 (\frac{1}{5} t^{5} + C_{3}) + 525 (\frac{1}{3} t^{3} + C_{4}) - 125 t + C_{5})$

Step 2.3.11.1.2

Combine $\frac{1}{5}$ and $t^{5}$ .

$s + C_{1} = 28 (343 (\frac{t^{7}}{7} + C_{2}) - 735 (\frac{t^{5}}{5} + C_{3}) + 525 (\frac{1}{3} t^{3} + C_{4}) - 125 t + C_{5})$

Step 2.3.11.1.3

Combine $\frac{1}{3}$ and $t^{3}$ .

$s + C_{1} = 28 (343 (\frac{t^{7}}{7} + C_{2}) - 735 (\frac{t^{5}}{5} + C_{3}) + 525 (\frac{t^{3}}{3} + C_{4}) - 125 t + C_{5})$

$s + C_{1} = 28 (343 (\frac{t^{7}}{7} + C_{2}) - 735 (\frac{t^{5}}{5} + C_{3}) + 525 (\frac{t^{3}}{3} + C_{4}) - 125 t + C_{5})$

Step 2.3.11.2

Simplify.

$s + C_{1} = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + C_{6}$

$s + C_{1} = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + C_{6}$

$s + C_{1} = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + C_{6}$

Step 2.4

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

$s = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + K$

$s = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $5$ for $s$ in $s = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + K$ .

$5 = 28 (49 \cdot 1^{7} - 147 \cdot 1^{5} + 175 \cdot 1^{3} - 125 \cdot 1) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $28 (49 \cdot 1^{7} - 147 \cdot 1^{5} + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$ .

$28 (49 \cdot 1^{7} - 147 \cdot 1^{5} + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$

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.

$28 (49 \cdot 1 - 147 \cdot 1^{5} + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$

Step 4.2.1.2

Multiply $49$ by $1$ .

$28 (49 - 147 \cdot 1^{5} + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$

Step 4.2.1.3

One to any power is one.

$28 (49 - 147 \cdot 1 + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$

Step 4.2.1.4

Multiply $- 147$ by $1$ .

$28 (49 - 147 + 175 \cdot 1^{3} - 125 \cdot 1) + K = 5$

Step 4.2.1.5

One to any power is one.

$28 (49 - 147 + 175 \cdot 1 - 125 \cdot 1) + K = 5$

Step 4.2.1.6

Multiply $175$ by $1$ .

$28 (49 - 147 + 175 - 125 \cdot 1) + K = 5$

Step 4.2.1.7

Multiply $- 125$ by $1$ .

$28 (49 - 147 + 175 - 125) + K = 5$

$28 (49 - 147 + 175 - 125) + K = 5$

Step 4.2.2

Subtract $147$ from $49$ .

$28 (- 98 + 175 - 125) + K = 5$

Step 4.2.3

Add $- 98$ and $175$ .

$28 (77 - 125) + K = 5$

Step 4.2.4

Subtract $125$ from $77$ .

$28 \cdot - 48 + K = 5$

Step 4.2.5

Multiply $28$ by $- 48$ .

$- 1344 + K = 5$

$- 1344 + K = 5$

Step 4.3

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

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Step 4.3.1

Add $1344$ to both sides of the equation.

$K = 5 + 1344$

Step 4.3.2

Add $5$ and $1344$ .

$K = 1349$

$K = 1349$

$K = 1349$

Step 5

Substitute $1349$ for $K$ in $s = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + K$ and simplify.

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Step 5.1

Substitute $1349$ for $K$ .

$s = 28 (49 t^{7} - 147 t^{5} + 175 t^{3} - 125 t) + 1349$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$s = 28 (49 t^{7}) + 28 (- 147 t^{5}) + 28 (175 t^{3}) + 28 (- 125 t) + 1349$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $49$ by $28$ .

$s = 1372 t^{7} + 28 (- 147 t^{5}) + 28 (175 t^{3}) + 28 (- 125 t) + 1349$

Step 5.2.2.2

Multiply $- 147$ by $28$ .

$s = 1372 t^{7} - 4116 t^{5} + 28 (175 t^{3}) + 28 (- 125 t) + 1349$

Step 5.2.2.3

Multiply $175$ by $28$ .

$s = 1372 t^{7} - 4116 t^{5} + 4900 t^{3} + 28 (- 125 t) + 1349$

Step 5.2.2.4

Multiply $- 125$ by $28$ .

$s = 1372 t^{7} - 4116 t^{5} + 4900 t^{3} - 3500 t + 1349$

$s = 1372 t^{7} - 4116 t^{5} + 4900 t^{3} - 3500 t + 1349$

$s = 1372 t^{7} - 4116 t^{5} + 4900 t^{3} - 3500 t + 1349$

$s = 1372 t^{7} - 4116 t^{5} + 4900 t^{3} - 3500 t + 1349$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$