Calculus Examples

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

Step 1

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

$\int \sqrt{{\sqrt{7}}^{2} - {(\sqrt{7} \sin (t))}^{2}} (\sqrt{7} \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

$\int \sqrt{{\sqrt{7}}^{2} - ({\sqrt{7}}^{2} \sin^{2} (t))} (\sqrt{7} \cos (t)) d t$

Step 2.1.2

Multiply by $1$ .

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

Step 2.1.3

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

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

Step 2.1.4

Factor ${\sqrt{7}}^{2}$ out of ${\sqrt{7}}^{2} \cdot 1 + {\sqrt{7}}^{2} (- (\sin^{2} (t)))$ .

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

Step 2.1.5

Apply pythagorean identity.

$\int \sqrt{{\sqrt{7}}^{2} \cdot \cos^{2} (t)} (\sqrt{7} \cos (t)) d t$

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

$\int \sqrt{7 \cdot \cos^{2} (t)} (\sqrt{7} \cos (t)) d t$

$\int \sqrt{7 \cos^{2} (t)} (\sqrt{7} \cos (t)) d t$

Step 2.1.7

Reorder $7$ and $\cos^{2} (t)$ .

$\int \sqrt{\cos^{2} (t) \cdot 7} (\sqrt{7} \cos (t)) d t$

Step 2.1.8

Pull terms out from under the radical.

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

Step 2.2

Simplify.

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Step 2.2.1

Raise $\cos (t)$ to the power of $1$ .

$\int \cos^{1} (t) \cos (t) \sqrt{7} \sqrt{7} d t$

Step 2.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int \cos^{1} (t) \cos^{1} (t) \sqrt{7} \sqrt{7} d t$

Step 2.2.3

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

$\int {\cos (t)}^{1 + 1} \sqrt{7} \sqrt{7} d t$

Step 2.2.4

Add $1$ and $1$ .

$\int \cos^{2} (t) \sqrt{7} \sqrt{7} d t$

Step 2.2.5

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

$\int \cos^{2} (t) ({\sqrt{7}}^{1} \sqrt{7}) d t$

Step 2.2.6

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

$\int \cos^{2} (t) ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) d t$

Step 2.2.7

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

$\int \cos^{2} (t) {\sqrt{7}}^{1 + 1} d t$

Step 2.2.8

Add $1$ and $1$ .

$\int \cos^{2} (t) {\sqrt{7}}^{2} d t$

Step 2.2.9

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

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Step 2.2.9.1

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

$\int \cos^{2} (t) {(7^{\frac{1}{2}})}^{2} d t$

Step 2.2.9.2

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

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

Step 2.2.9.3

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

$\int \cos^{2} (t) \cdot 7^{\frac{2}{2}} d t$

Step 2.2.9.4

Cancel the common factor of $2$ .

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Step 2.2.9.4.1

Cancel the common factor.

$\int \cos^{2} (t) \cdot 7^{\frac{2}{2}} d t$

Step 2.2.9.4.2

Rewrite the expression.

$\int \cos^{2} (t) \cdot 7^{1} d t$

Step 2.2.9.5

Evaluate the exponent.

$\int \cos^{2} (t) \cdot 7 d t$

Step 2.2.10

Move $7$ to the left of $\cos^{2} (t)$ .

$\int 7 \cos^{2} (t) d t$

Step 3

Since $7$ is constant with respect to $t$ , move $7$ out of the integral.

$7 \int \cos^{2} (t) d t$

Step 4

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$7 \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$7 (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Combine $\frac{1}{2}$ and $7$ .

$\frac{7}{2} \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{7}{2} (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{7}{2} (t + C + \int \cos (2 t) d t)$

Step 9

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 9.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 9.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 9.1.2

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

$2 \frac{d}{d t} [t]$

Step 9.1.3

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

$2 \cdot 1$

Step 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

$\frac{7}{2} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{7}{2} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{7}{2} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{7}{2} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$\frac{7}{2} (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (u)) + C$

Step 14.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 t)) + C$

Step 14.3

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x}{\sqrt{7}}))) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}))) + C$

Step 15.1.2

Combine and simplify the denominator.

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Step 15.1.2.1

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{\sqrt{7} \sqrt{7}}))) + C$

Step 15.1.2.2

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}))) + C$

Step 15.1.2.3

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}))) + C$

Step 15.1.2.4

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1 + 1}}))) + C$

Step 15.1.2.5

Add $1$ and $1$ .

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{2}}))) + C$

Step 15.1.2.6

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

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Step 15.1.2.6.1

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}))) + C$

Step 15.1.2.6.2

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}))) + C$

Step 15.1.2.6.3

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

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{7^{\frac{2}{2}}}))) + C$

Step 15.1.2.6.4

Cancel the common factor of $2$ .

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Step 15.1.2.6.4.1

Cancel the common factor.

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{7^{\frac{2}{2}}}))) + C$

Step 15.1.2.6.4.2

Rewrite the expression.

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{7^{1}}))) + C$

Step 15.1.2.6.5

Evaluate the exponent.

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{1}{2} \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))) + C$

Step 15.1.3

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))$ .

$\frac{7}{2} (\arcsin (\frac{x}{\sqrt{7}}) + \frac{\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{7}{2} \arcsin (\frac{x}{\sqrt{7}}) + \frac{7}{2} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2} + C$

Step 15.3

Combine $\frac{7}{2}$ and $\arcsin (\frac{x}{\sqrt{7}})$ .

$\frac{7 \arcsin (\frac{x}{\sqrt{7}})}{2} + \frac{7}{2} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2} + C$

Step 15.4

Multiply $\frac{7}{2} \cdot \frac{\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2}$ .

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Step 15.4.1

Multiply $\frac{7}{2}$ by $\frac{\sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2}$ .

$\frac{7 \arcsin (\frac{x}{\sqrt{7}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{2 \cdot 2} + C$

Step 15.4.2

Multiply $2$ by $2$ .

$\frac{7 \arcsin (\frac{x}{\sqrt{7}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16

Simplify.

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Step 16.1

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

$\frac{7 \arcsin (\frac{x}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2

Combine and simplify the denominator.

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Step 16.2.1

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{\sqrt{7} \sqrt{7}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.2

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.3

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.4

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{1 + 1}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.5

Add $1$ and $1$ .

$\frac{7 \arcsin (\frac{x \sqrt{7}}{{\sqrt{7}}^{2}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6

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

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Step 16.2.6.1

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6.2

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{7^{\frac{1}{2} \cdot 2}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6.3

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

$\frac{7 \arcsin (\frac{x \sqrt{7}}{7^{\frac{2}{2}}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6.4

Cancel the common factor of $2$ .

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Step 16.2.6.4.1

Cancel the common factor.

$\frac{7 \arcsin (\frac{x \sqrt{7}}{7^{\frac{2}{2}}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6.4.2

Rewrite the expression.

$\frac{7 \arcsin (\frac{x \sqrt{7}}{7^{1}})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.2.6.5

Evaluate the exponent.

$\frac{7 \arcsin (\frac{x \sqrt{7}}{7})}{2} + \frac{7 \sin (2 \arcsin (\frac{x \sqrt{7}}{7}))}{4} + C$

Step 16.3

Reorder terms.

$\frac{7}{2} \arcsin (\frac{\sqrt{7}}{7} x) + \frac{7}{4} \sin (2 \arcsin (\frac{\sqrt{7}}{7} x)) + C$