Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{3}} (3 \tan (x) \sec (x) + 2 \cos (x)) d x$

Step 1

Remove parentheses.

$\int_{\frac{π}{4}}^{\frac{π}{3}} 3 \tan (x) \sec (x) + 2 \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{\frac{π}{4}}^{\frac{π}{3}} 3 \tan (x) \sec (x) d x + \int_{\frac{π}{4}}^{\frac{π}{3}} 2 \cos (x) d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{\frac{π}{4}}^{\frac{π}{3}} \tan (x) \sec (x) d x + \int_{\frac{π}{4}}^{\frac{π}{3}} 2 \cos (x) d x$

Step 4

Since the derivative of $\sec (x)$ is $\tan (x) \sec (x)$ , the integral of $\tan (x) \sec (x)$ is $\sec (x)$ .

$3 (\sec (x)]_{\frac{π}{4}}^{\frac{π}{3}}) + \int_{\frac{π}{4}}^{\frac{π}{3}} 2 \cos (x) d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$3 (\sec (x)]_{\frac{π}{4}}^{\frac{π}{3}}) + 2 \int_{\frac{π}{4}}^{\frac{π}{3}} \cos (x) d x$

Step 6

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

$3 (\sec (x)]_{\frac{π}{4}}^{\frac{π}{3}}) + 2 (\sin (x)]_{\frac{π}{4}}^{\frac{π}{3}})$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

Evaluate $\sec (x)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$3 ((\sec (\frac{π}{3})) - \sec (\frac{π}{4})) + 2 (\sin (x)]_{\frac{π}{4}}^{\frac{π}{3}})$

Step 7.1.2

Evaluate $\sin (x)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$3 (\sec (\frac{π}{3}) - \sec (\frac{π}{4})) + 2 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2

Simplify.

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Step 7.2.1

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$3 (2 - \sec (\frac{π}{4})) + 2 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2.2

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$3 (2 - \frac{2}{\sqrt{2}}) + 2 (\sin (\frac{π}{3}) - \sin (\frac{π}{4}))$

Step 7.2.3

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$3 (2 - \frac{2}{\sqrt{2}}) + 2 (\frac{\sqrt{3}}{2} - \sin (\frac{π}{4}))$

Step 7.2.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$3 (2 - \frac{2}{\sqrt{2}}) + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3

Simplify.

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Step 7.3.1

Apply the distributive property.

$3 \cdot 2 + 3 (- \frac{2}{\sqrt{2}}) + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.2

Multiply $3$ by $2$ .

$6 + 3 (- \frac{2}{\sqrt{2}}) + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.3

Multiply $3 (- \frac{2}{\sqrt{2}})$ .

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Step 7.3.3.1

Multiply $- 1$ by $3$ .

$6 - 3 \frac{2}{\sqrt{2}} + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.3.2

Combine $- 3$ and $\frac{2}{\sqrt{2}}$ .

$6 + \frac{- 3 \cdot 2}{\sqrt{2}} + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.3.3

Multiply $- 3$ by $2$ .

$6 + \frac{- 6}{\sqrt{2}} + 2 (\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2})$

Step 7.3.4

Apply the distributive property.

$6 + \frac{- 6}{\sqrt{2}} + 2 \frac{\sqrt{3}}{2} + 2 (- \frac{\sqrt{2}}{2})$

Step 7.3.5

Cancel the common factor of $2$ .

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Step 7.3.5.1

Cancel the common factor.

$6 + \frac{- 6}{\sqrt{2}} + 2 \frac{\sqrt{3}}{2} + 2 (- \frac{\sqrt{2}}{2})$

Step 7.3.5.2

Rewrite the expression.

$6 + \frac{- 6}{\sqrt{2}} + \sqrt{3} + 2 (- \frac{\sqrt{2}}{2})$

Step 7.3.6

Cancel the common factor of $2$ .

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Step 7.3.6.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$6 + \frac{- 6}{\sqrt{2}} + \sqrt{3} + 2 \frac{- \sqrt{2}}{2}$

Step 7.3.6.2

Cancel the common factor.

$6 + \frac{- 6}{\sqrt{2}} + \sqrt{3} + 2 \frac{- \sqrt{2}}{2}$

Step 7.3.6.3

Rewrite the expression.

$6 + \frac{- 6}{\sqrt{2}} + \sqrt{3} - \sqrt{2}$

Step 7.3.7

Move the negative in front of the fraction.

$6 - \frac{6}{\sqrt{2}} + \sqrt{3} - \sqrt{2}$

Step 7.4

Simplify.

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Step 7.4.1

Simplify each term.

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Step 7.4.1.1

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$6 - (\frac{6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2

Combine and simplify the denominator.

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Step 7.4.1.2.1

Multiply $\frac{6}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$6 - \frac{6 \sqrt{2}}{\sqrt{2} \sqrt{2}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$6 - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$6 - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.4

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

$6 - \frac{6 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.5

Add $1$ and $1$ .

$6 - \frac{6 \sqrt{2}}{{\sqrt{2}}^{2}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6

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

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Step 7.4.1.2.6.1

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

$6 - \frac{6 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6.2

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

$6 - \frac{6 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6.3

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

$6 - \frac{6 \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6.4

Cancel the common factor of $2$ .

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Step 7.4.1.2.6.4.1

Cancel the common factor.

$6 - \frac{6 \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6.4.2

Rewrite the expression.

$6 - \frac{6 \sqrt{2}}{2^{1}} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.2.6.5

Evaluate the exponent.

$6 - \frac{6 \sqrt{2}}{2} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.3

Cancel the common factor of $6$ and $2$ .

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Step 7.4.1.3.1

Factor $2$ out of $6 \sqrt{2}$ .

$6 - \frac{2 (3 \sqrt{2})}{2} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.3.2

Cancel the common factors.

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Step 7.4.1.3.2.1

Factor $2$ out of $2$ .

$6 - \frac{2 (3 \sqrt{2})}{2 (1)} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.3.2.2

Cancel the common factor.

$6 - \frac{2 (3 \sqrt{2})}{2 \cdot 1} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.3.2.3

Rewrite the expression.

$6 - \frac{3 \sqrt{2}}{1} + \sqrt{3} - \sqrt{2}$

Step 7.4.1.3.2.4

Divide $3 \sqrt{2}$ by $1$ .

$6 - (3 \sqrt{2}) + \sqrt{3} - \sqrt{2}$

Step 7.4.1.4

Multiply $3$ by $- 1$ .

$6 - 3 \sqrt{2} + \sqrt{3} - \sqrt{2}$

Step 7.4.2

Subtract $\sqrt{2}$ from $- 3 \sqrt{2}$ .

$6 - 4 \sqrt{2} + \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 - 4 \sqrt{2} + \sqrt{3}$

Decimal Form:

$2.07519655 \dots$