Calculus Examples

$\frac{d s}{d t} = 20 t {(5 t^{2} - 3)}^{3}$ , $s (1) = 7$

Step 1

Rewrite the equation.

$d s = 20 t {(5 t^{2} - 3)}^{3} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 20 t {(5 t^{2} - 3)}^{3} d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 20 t {(5 t^{2} - 3)}^{3} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $20$ is constant with respect to $t$ , move $20$ out of the integral.

$s + C_{1} = 20 \int t {(5 t^{2} - 3)}^{3} d t$

Step 2.3.2

Let $u = 5 t^{2} - 3$ . Then $d u = 10 t d t$ , so $\frac{1}{10} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 5 t^{2} - 3$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $5 t^{2} - 3$ .

$\frac{d}{d t} [5 t^{2} - 3]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $5 t^{2} - 3$ with respect to $t$ is $\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [- 3]$ .

$\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [- 3]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d t} [5 t^{2}]$ .

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Step 2.3.2.1.3.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{2}$ with respect to $t$ is $5 \frac{d}{d t} [t^{2}]$ .

$5 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 3]$

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 $n = 2$ .

$5 (2 t) + \frac{d}{d t} [- 3]$

Step 2.3.2.1.3.3

Multiply $2$ by $5$ .

$10 t + \frac{d}{d t} [- 3]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3$ with respect to $t$ is $0$ .

$10 t + 0$

Step 2.3.2.1.4.2

Add $10 t$ and $0$ .

$10 t$

Step 2.3.2.2

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

$s + C_{1} = 20 \int u^{3} \frac{1}{10} d u$

Step 2.3.3

Combine $u^{3}$ and $\frac{1}{10}$ .

$s + C_{1} = 20 \int \frac{u^{3}}{10} d u$

Step 2.3.4

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$s + C_{1} = 20 (\frac{1}{10} \int u^{3} d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{10}$ and $20$ .

$s + C_{1} = \frac{20}{10} \int u^{3} d u$

Step 2.3.5.2

Cancel the common factor of $20$ and $10$ .

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Step 2.3.5.2.1

Factor $10$ out of $20$ .

$s + C_{1} = \frac{10 \cdot 2}{10} \int u^{3} d u$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $10$ out of $10$ .

$s + C_{1} = \frac{10 \cdot 2}{10 (1)} \int u^{3} d u$

Step 2.3.5.2.2.2

Cancel the common factor.

$s + C_{1} = \frac{10 \cdot 2}{10 \cdot 1} \int u^{3} d u$

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 $5 t^{2} - 3$ .

$s + C_{1} = \frac{1}{2} {(5 t^{2} - 3)}^{4} + C_{2}$

Step 2.4

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

$s = \frac{1}{2} {(5 t^{2} - 3)}^{4} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $7$ for $s$ in $s = \frac{1}{2} {(5 t^{2} - 3)}^{4} + K$ .

$7 = \frac{1}{2} {(5 \cdot 1^{2} - 3)}^{4} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{2} \cdot {(5 \cdot 1^{2} - 3)}^{4} + K = 7$ .

$\frac{1}{2} \cdot {(5 \cdot 1^{2} - 3)}^{4} + K = 7$

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.

$\frac{1}{2} \cdot {(5 \cdot 1 - 3)}^{4} + K = 7$

Step 4.2.1.2

Multiply $5$ by $1$ .

$\frac{1}{2} \cdot {(5 - 3)}^{4} + K = 7$

Step 4.2.2

Subtract $3$ from $5$ .

$\frac{1}{2} \cdot 2^{4} + K = 7$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

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

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

Step 4.2.3.2

Cancel the common factor.

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

Step 4.2.3.3

Rewrite the expression.

$2^{3} + K = 7$

Step 4.2.4

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

$8 + K = 7$

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.

$K = 7 - 8$

Step 4.3.2

Subtract $8$ from $7$ .

$K = - 1$

Step 5

Substitute $- 1$ for $K$ in $s = \frac{1}{2} {(5 t^{2} - 3)}^{4} + K$ and simplify.

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Step 5.1

Substitute $- 1$ for $K$ .

$s = \frac{1}{2} {(5 t^{2} - 3)}^{4} - 1$

Step 5.2

Simplify each term.

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Step 5.2.1

Use the Binomial Theorem.

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Apply the product rule to $5 t^{2}$ .

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

Step 5.2.2.2

Raise $5$ to the power of $4$ .

$s = \frac{1}{2} (625 {(t^{2})}^{4} + 4 {(5 t^{2})}^{3} \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

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, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (625 t^{2 \cdot 4} + 4 {(5 t^{2})}^{3} \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.3.2

Multiply $2$ by $4$ .

$s = \frac{1}{2} (625 t^{8} + 4 {(5 t^{2})}^{3} \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.4

Apply the product rule to $5 t^{2}$ .

$s = \frac{1}{2} (625 t^{8} + 4 (5^{3} {(t^{2})}^{3}) \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.5

Raise $5$ to the power of $3$ .

$s = \frac{1}{2} (625 t^{8} + 4 (125 {(t^{2})}^{3}) \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

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, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (625 t^{8} + 4 (125 t^{2 \cdot 3}) \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.6.2

Multiply $2$ by $3$ .

$s = \frac{1}{2} (625 t^{8} + 4 (125 t^{6}) \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.7

Multiply $125$ by $4$ .

$s = \frac{1}{2} (625 t^{8} + 500 t^{6} \cdot - 3 + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.8

Multiply $- 3$ by $500$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 6 {(5 t^{2})}^{2} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.9

Apply the product rule to $5 t^{2}$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 6 (5^{2} {(t^{2})}^{2}) {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.10

Raise $5$ to the power of $2$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 6 (25 {(t^{2})}^{2}) {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

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, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 6 (25 t^{2 \cdot 2}) {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.11.2

Multiply $2$ by $2$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 6 (25 t^{4}) {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.12

Multiply $25$ by $6$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 150 t^{4} {(- 3)}^{2} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.13

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

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 150 t^{4} \cdot 9 + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.14

Multiply $9$ by $150$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 1350 t^{4} + 4 (5 t^{2}) {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.15

Multiply $5$ by $4$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 1350 t^{4} + 20 t^{2} {(- 3)}^{3} + {(- 3)}^{4}) - 1$

Step 5.2.2.16

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

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 1350 t^{4} + 20 t^{2} \cdot - 27 + {(- 3)}^{4}) - 1$

Step 5.2.2.17

Multiply $- 27$ by $20$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 1350 t^{4} - 540 t^{2} + {(- 3)}^{4}) - 1$

Step 5.2.2.18

Raise $- 3$ to the power of $4$ .

$s = \frac{1}{2} (625 t^{8} - 1500 t^{6} + 1350 t^{4} - 540 t^{2} + 81) - 1$

Step 5.2.3

Apply the distributive property.

$s = \frac{1}{2} (625 t^{8}) + \frac{1}{2} (- 1500 t^{6}) + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Multiply $\frac{1}{2} (625 t^{8})$ .

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Step 5.2.4.1.1

Combine $625$ and $\frac{1}{2}$ .

$s = \frac{625}{2} t^{8} + \frac{1}{2} (- 1500 t^{6}) + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.1.2

Combine $\frac{625}{2}$ and $t^{8}$ .

$s = \frac{625 t^{8}}{2} + \frac{1}{2} (- 1500 t^{6}) + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

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 $- 1500 t^{6}$ .

$s = \frac{625 t^{8}}{2} + \frac{1}{2} (2 (- 750 t^{6})) + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.2.2

Cancel the common factor.

$s = \frac{625 t^{8}}{2} + \frac{1}{2} (2 (- 750 t^{6})) + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.2.3

Rewrite the expression.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + \frac{1}{2} (1350 t^{4}) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

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 $1350 t^{4}$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + \frac{1}{2} (2 (675 t^{4})) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.3.2

Cancel the common factor.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + \frac{1}{2} (2 (675 t^{4})) + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.3.3

Rewrite the expression.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} + \frac{1}{2} (- 540 t^{2}) + \frac{1}{2} \cdot 81 - 1$

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 $- 540 t^{2}$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} + \frac{1}{2} (2 (- 270 t^{2})) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.4.2

Cancel the common factor.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} + \frac{1}{2} (2 (- 270 t^{2})) + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.4.3

Rewrite the expression.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{1}{2} \cdot 81 - 1$

Step 5.2.4.5

Combine $\frac{1}{2}$ and $81$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{81}{2} - 1$

Step 5.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{81}{2} - 1 \cdot \frac{2}{2}$

Step 5.4

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

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{81}{2} + \frac{- 1 \cdot 2}{2}$

Step 5.5

Combine the numerators over the common denominator.

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{81 - 1 \cdot 2}{2}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 1$ by $2$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{81 - 2}{2}$

Step 5.6.2

Subtract $2$ from $81$ .

$s = \frac{625 t^{8}}{2} - 750 t^{6} + 675 t^{4} - 270 t^{2} + \frac{79}{2}$