Calculus Examples

$2 \sqrt{4902525} \frac{d c}{d t} = 2 (1755) (540) + 2 (1350) (600)$

Step 1

Rewrite the equation.

$2 \sqrt{4902525} d c = (2 (1755) (540) + 2 (1350) (600)) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 \sqrt{4902525} d c = \int 2 (1755) (540) + 2 (1350) (600) d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply the constant rule.

$2 \sqrt{4902525} c + C_{1} = \int 2 (1755) (540) + 2 (1350) (600) d t$

Step 2.2.2

Simplify the answer.

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Step 2.2.2.1

Rewrite $2 \sqrt{4902525} c + C_{1}$ as $2 (135 \sqrt{269}) c + C_{1}$ .

$2 (135 \sqrt{269}) c + C_{1} = \int 2 (1755) (540) + 2 (1350) (600) d t$

Step 2.2.2.2

Multiply $135$ by $2$ .

$270 \sqrt{269} c + C_{1} = \int 2 (1755) (540) + 2 (1350) (600) d t$

Step 2.2.3

Reorder terms.

$270 c \sqrt{269} + C_{1} = \int 2 (1755) (540) + 2 (1350) (600) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

Multiply $2$ by $1755$ .

$270 c \sqrt{269} + C_{1} = \int 3510 \cdot 540 + 2 (1350) (600) d t$

Step 2.3.1.2

Multiply $3510$ by $540$ .

$270 c \sqrt{269} + C_{1} = \int 1895400 + 2 (1350) (600) d t$

Step 2.3.1.3

Multiply $2$ by $1350$ .

$270 c \sqrt{269} + C_{1} = \int 1895400 + 2700 \cdot 600 d t$

Step 2.3.1.4

Multiply $2700$ by $600$ .

$270 c \sqrt{269} + C_{1} = \int 1895400 + 1620000 d t$

Step 2.3.1.5

Add $1895400$ and $1620000$ .

$270 c \sqrt{269} + C_{1} = \int 3515400 d t$

Step 2.3.2

Apply the constant rule.

$270 c \sqrt{269} + C_{1} = 3515400 t + C_{2}$

Step 2.4

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

$270 c \sqrt{269} = 3515400 t + K$

Step 3

Divide each term in $270 c \sqrt{269} = 3515400 t + K$ by $270 \sqrt{269}$ and simplify.

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Step 3.1

Divide each term in $270 c \sqrt{269} = 3515400 t + K$ by $270 \sqrt{269}$ .

$\frac{270 c \sqrt{269}}{270 \sqrt{269}} = \frac{3515400 t}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $270$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{270 c \sqrt{269}}{270 \sqrt{269}} = \frac{3515400 t}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.2.1.2

Rewrite the expression.

$\frac{c \sqrt{269}}{\sqrt{269}} = \frac{3515400 t}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.2.2

Cancel the common factor of $\sqrt{269}$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{c \sqrt{269}}{\sqrt{269}} = \frac{3515400 t}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.2.2.2

Divide $c$ by $1$ .

$c = \frac{3515400 t}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $3515400$ and $270$ .

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Step 3.3.1.1.1

Factor $270$ out of $3515400 t$ .

$c = \frac{270 (13020 t)}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $270$ out of $270 \sqrt{269}$ .

$c = \frac{270 (13020 t)}{270 (\sqrt{269})} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.1.2.2

Cancel the common factor.

$c = \frac{270 (13020 t)}{270 \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.1.2.3

Rewrite the expression.

$c = \frac{13020 t}{\sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.2

Multiply $\frac{13020 t}{\sqrt{269}}$ by $\frac{\sqrt{269}}{\sqrt{269}}$ .

$c = \frac{13020 t}{\sqrt{269}} \cdot \frac{\sqrt{269}}{\sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3

Combine and simplify the denominator.

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Step 3.3.1.3.1

Multiply $\frac{13020 t}{\sqrt{269}}$ by $\frac{\sqrt{269}}{\sqrt{269}}$ .

$c = \frac{13020 t \sqrt{269}}{\sqrt{269} \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.2

Raise $\sqrt{269}$ to the power of $1$ .

$c = \frac{13020 t \sqrt{269}}{{\sqrt{269}}^{1} \sqrt{269}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.3

Raise $\sqrt{269}$ to the power of $1$ .

$c = \frac{13020 t \sqrt{269}}{{\sqrt{269}}^{1} {\sqrt{269}}^{1}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.4

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

$c = \frac{13020 t \sqrt{269}}{{\sqrt{269}}^{1 + 1}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.5

Add $1$ and $1$ .

$c = \frac{13020 t \sqrt{269}}{{\sqrt{269}}^{2}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6

Rewrite ${\sqrt{269}}^{2}$ as $269$ .

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Step 3.3.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{269}$ as $269^{\frac{1}{2}}$ .

$c = \frac{13020 t \sqrt{269}}{{(269^{\frac{1}{2}})}^{2}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6.2

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

$c = \frac{13020 t \sqrt{269}}{269^{\frac{1}{2} \cdot 2}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6.3

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

$c = \frac{13020 t \sqrt{269}}{269^{\frac{2}{2}}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.6.4.1

Cancel the common factor.

$c = \frac{13020 t \sqrt{269}}{269^{\frac{2}{2}}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6.4.2

Rewrite the expression.

$c = \frac{13020 t \sqrt{269}}{269^{1}} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.3.6.5

Evaluate the exponent.

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K}{270 \sqrt{269}}$

Step 3.3.1.4

Multiply $\frac{K}{270 \sqrt{269}}$ by $\frac{\sqrt{269}}{\sqrt{269}}$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K}{270 \sqrt{269}} \cdot \frac{\sqrt{269}}{\sqrt{269}}$

Step 3.3.1.5

Combine and simplify the denominator.

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Step 3.3.1.5.1

Multiply $\frac{K}{270 \sqrt{269}}$ by $\frac{\sqrt{269}}{\sqrt{269}}$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \sqrt{269} \sqrt{269}}$

Step 3.3.1.5.2

Move $\sqrt{269}$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 (\sqrt{269} \sqrt{269})}$

Step 3.3.1.5.3

Raise $\sqrt{269}$ to the power of $1$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 ({\sqrt{269}}^{1} \sqrt{269})}$

Step 3.3.1.5.4

Raise $\sqrt{269}$ to the power of $1$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 ({\sqrt{269}}^{1} {\sqrt{269}}^{1})}$

Step 3.3.1.5.5

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

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 {\sqrt{269}}^{1 + 1}}$

Step 3.3.1.5.6

Add $1$ and $1$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 {\sqrt{269}}^{2}}$

Step 3.3.1.5.7

Rewrite ${\sqrt{269}}^{2}$ as $269$ .

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Step 3.3.1.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{269}$ as $269^{\frac{1}{2}}$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 {(269^{\frac{1}{2}})}^{2}}$

Step 3.3.1.5.7.2

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

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \cdot 269^{\frac{1}{2} \cdot 2}}$

Step 3.3.1.5.7.3

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

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \cdot 269^{\frac{2}{2}}}$

Step 3.3.1.5.7.4

Cancel the common factor of $2$ .

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Step 3.3.1.5.7.4.1

Cancel the common factor.

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \cdot 269^{\frac{2}{2}}}$

Step 3.3.1.5.7.4.2

Rewrite the expression.

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \cdot 269^{1}}$

Step 3.3.1.5.7.5

Evaluate the exponent.

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{270 \cdot 269}$

Step 3.3.1.6

Multiply $270$ by $269$ .

$c = \frac{13020 t \sqrt{269}}{269} + \frac{K \sqrt{269}}{72630}$

Step 4

Simplify the constant of integration.

$c = \frac{13020 t \sqrt{269}}{269} + K$