Calculus Examples

$\int \frac{1}{\sqrt{5 - 7 x^{2}}} d x$

Step 1

Let $x = \frac{\sqrt{5}}{\sqrt{7}} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{35} \cos (t)}{7} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{35} \cos (t)}{7}$ is positive.

$\int \frac{1}{\sqrt{{\sqrt{5}}^{2} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{\sqrt{5}}^{2} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \frac{1}{\sqrt{{(5^{\frac{1}{2}})}^{2} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.1.2

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

$\int \frac{1}{\sqrt{5^{\frac{1}{2} \cdot 2} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.1.3

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

$\int \frac{1}{\sqrt{5^{\frac{2}{2}} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.1.4.1

Cancel the common factor.

$\int \frac{1}{\sqrt{5^{\frac{2}{2}} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.1.4.2

Rewrite the expression.

$\int \frac{1}{\sqrt{5^{1} - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.1.5

Evaluate the exponent.

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.2

Multiply $\frac{\sqrt{5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3

Combine and simplify the denominator.

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Step 2.1.1.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.4

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

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1 + 1}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.5

Add $1$ and $1$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{2}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6.2

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

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6.3

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

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.6.4.1

Cancel the common factor.

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6.4.2

Rewrite the expression.

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{7^{1}} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.3.6.5

Evaluate the exponent.

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5} \sqrt{7}}{7} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.4

Simplify the numerator.

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Step 2.1.1.4.1

Combine using the product rule for radicals.

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{5 \cdot 7}}{7} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.4.2

Multiply $5$ by $7$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{35}}{7} \sin (t))}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.5

Combine $\frac{\sqrt{35}}{7}$ and $\sin (t)$ .

$\int \frac{1}{\sqrt{5 - 7 {(\frac{\sqrt{35} \sin (t)}{7})}^{2}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.6.1

Apply the product rule to $\frac{\sqrt{35} \sin (t)}{7}$ .

$\int \frac{1}{\sqrt{5 - 7 \frac{{(\sqrt{35} \sin (t))}^{2}}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.6.2

Apply the product rule to $\sqrt{35} \sin (t)$ .

$\int \frac{1}{\sqrt{5 - 7 \frac{{\sqrt{35}}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 2.1.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\int \frac{1}{\sqrt{5 - 7 \frac{{(35^{\frac{1}{2}})}^{2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7.2

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

$\int \frac{1}{\sqrt{5 - 7 \frac{35^{\frac{1}{2} \cdot 2} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7.3

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

$\int \frac{1}{\sqrt{5 - 7 \frac{35^{\frac{2}{2}} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.1.7.4.1

Cancel the common factor.

$\int \frac{1}{\sqrt{5 - 7 \frac{35^{\frac{2}{2}} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7.4.2

Rewrite the expression.

$\int \frac{1}{\sqrt{5 - 7 \frac{35^{1} \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.7.5

Evaluate the exponent.

$\int \frac{1}{\sqrt{5 - 7 \frac{35 \sin^{2} (t)}{7^{2}}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.8

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

$\int \frac{1}{\sqrt{5 - 7 \frac{35 \sin^{2} (t)}{49}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.9

Cancel the common factor of $7$ .

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Step 2.1.1.9.1

Factor $7$ out of $- 7$ .

$\int \frac{1}{\sqrt{5 + 7 (- 1) \frac{35 \sin^{2} (t)}{49}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.9.2

Factor $7$ out of $49$ .

$\int \frac{1}{\sqrt{5 + 7 \cdot - 1 \frac{35 \sin^{2} (t)}{7 \cdot 7}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.9.3

Cancel the common factor.

$\int \frac{1}{\sqrt{5 + 7 \cdot - 1 \frac{35 \sin^{2} (t)}{7 \cdot 7}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.9.4

Rewrite the expression.

$\int \frac{1}{\sqrt{5 - 1 \frac{35 \sin^{2} (t)}{7}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.10

Cancel the common factor of $35$ and $7$ .

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Step 2.1.1.10.1

Factor $7$ out of $35 \sin^{2} (t)$ .

$\int \frac{1}{\sqrt{5 - 1 \frac{7 (5 \sin^{2} (t))}{7}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.10.2

Cancel the common factors.

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Step 2.1.1.10.2.1

Factor $7$ out of $7$ .

$\int \frac{1}{\sqrt{5 - 1 \frac{7 (5 \sin^{2} (t))}{7 (1)}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.10.2.2

Cancel the common factor.

$\int \frac{1}{\sqrt{5 - 1 \frac{7 (5 \sin^{2} (t))}{7 \cdot 1}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.10.2.3

Rewrite the expression.

$\int \frac{1}{\sqrt{5 - 1 \frac{5 \sin^{2} (t)}{1}}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.10.2.4

Divide $5 \sin^{2} (t)$ by $1$ .

$\int \frac{1}{\sqrt{5 - 1 (5 \sin^{2} (t))}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.1.11

Multiply $5$ by $- 1$ .

$\int \frac{1}{\sqrt{5 - 5 \sin^{2} (t)}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.2

Factor $5$ out of $5$ .

$\int \frac{1}{\sqrt{5 (1) - 5 \sin^{2} (t)}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.3

Factor $5$ out of $- 5 \sin^{2} (t)$ .

$\int \frac{1}{\sqrt{5 (1) + 5 (- \sin^{2} (t))}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.4

Factor $5$ out of $5 (1) + 5 (- \sin^{2} (t))$ .

$\int \frac{1}{\sqrt{5 (1 - \sin^{2} (t))}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{1}{\sqrt{5 \cos^{2} (t)}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.6

Reorder $5$ and $\cos^{2} (t)$ .

$\int \frac{1}{\sqrt{\cos^{2} (t) \cdot 5}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.1.7

Pull terms out from under the radical.

$\int \frac{1}{\cos (t) \sqrt{5}} \cdot \frac{\sqrt{35} \cos (t)}{7} d t$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $\frac{1}{\cos (t) \sqrt{5}}$ by $\frac{\sqrt{35} \cos (t)}{7}$ .

$\int \frac{\sqrt{35} \cos (t)}{\cos (t) \sqrt{5} \cdot 7} d t$

Step 2.2.2

Move $7$ to the left of $\cos (t) \sqrt{5}$ .

$\int \frac{\sqrt{35} \cos (t)}{7 \cdot (\cos (t) \sqrt{5})} d t$

Step 2.2.3

Cancel the common factor of $\cos (t)$ .

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Step 2.2.3.1

Cancel the common factor.

$\int \frac{\sqrt{35} \cos (t)}{7 \cos (t) \sqrt{5}} d t$

Step 2.2.3.2

Rewrite the expression.

$\int \frac{\sqrt{35}}{7 \sqrt{5}} d t$

Step 3

Apply the constant rule.

$\frac{\sqrt{35}}{7 \sqrt{5}} t + C$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{\sqrt{35}}{7 \sqrt{5}} t + C$ as $\frac{\sqrt{\frac{35}{5}}}{7} t + C$ .

$\frac{\sqrt{\frac{35}{5}}}{7} t + C$

Step 4.2

Rewrite $\frac{\sqrt{\frac{35}{5}}}{7} t + C$ as $\frac{\sqrt{7}}{7} t + C$ .

$\frac{\sqrt{7}}{7} t + C$

Step 4.3

Replace all occurrences of $t$ with $\arcsin (\frac{\sqrt{7} x}{\sqrt{5}})$ .

$\frac{\sqrt{7}}{7} \arcsin (\frac{\sqrt{7} x}{\sqrt{5}}) + C$

Step 4.4

Reorder terms.

$\frac{\sqrt{7}}{7} \arcsin (\frac{\sqrt{7}}{\sqrt{5}} x) + C$