Calculus Examples

$\int \frac{x \sec (x) \tan (x) + 3 x^{2} + \sqrt{x}}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int \frac{x \sec (x) \tan (x)}{x} + \frac{3 x^{2}}{x} + \frac{\sqrt{x}}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{x \sec (x) \tan (x)}{x} d x + \int \frac{3 x^{2}}{x} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

$\int \frac{x \sec (x) \tan (x)}{x} d x + \int \frac{3 x^{2}}{x} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.1.2

Divide $\sec (x) \tan (x)$ by $1$ .

$\int \sec (x) \tan (x) d x + \int \frac{3 x^{2}}{x} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.2.1

Factor $x$ out of $3 x^{2}$ .

$\int \sec (x) \tan (x) d x + \int \frac{x (3 x)}{x} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Raise $x$ to the power of $1$ .

$\int \sec (x) \tan (x) d x + \int \frac{x (3 x)}{x^{1}} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2.2.2

Factor $x$ out of $x^{1}$ .

$\int \sec (x) \tan (x) d x + \int \frac{x (3 x)}{x \cdot 1} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2.2.3

Cancel the common factor.

$\int \sec (x) \tan (x) d x + \int \frac{x (3 x)}{x \cdot 1} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2.2.4

Rewrite the expression.

$\int \sec (x) \tan (x) d x + \int \frac{3 x}{1} d x + \int \frac{\sqrt{x}}{x} d x$

Step 3.2.2.5

Divide $3 x$ by $1$ .

$\int \sec (x) \tan (x) d x + \int 3 x d x + \int \frac{\sqrt{x}}{x} d x$

Step 4

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

$\sec (x) + C + \int 3 x d x + \int \frac{\sqrt{x}}{x} d x$

Step 5

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

$\sec (x) + C + 3 \int x d x + \int \frac{\sqrt{x}}{x} d x$

Step 6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\sec (x) + C + 3 (\frac{1}{2} x^{2} + C) + \int \frac{\sqrt{x}}{x} d x$

Step 7

Simplify the expression.

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Step 7.1

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

$\sec (x) + C + 3 (\frac{1}{2} x^{2} + C) + \int \frac{x^{\frac{1}{2}}}{x} d x$

Step 7.2

Simplify.

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Step 7.2.1

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

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{x^{\frac{1}{2}}}{x} d x$

Step 7.2.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x \cdot x^{- \frac{1}{2}}} d x$

Step 7.2.3

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.2.3.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 7.2.3.1.1

Raise $x$ to the power of $1$ .

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x^{1} x^{- \frac{1}{2}}} d x$

Step 7.2.3.1.2

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

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x^{1 - \frac{1}{2}}} d x$

Step 7.2.3.2

Write $1$ as a fraction with a common denominator.

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x^{\frac{2}{2} - \frac{1}{2}}} d x$

Step 7.2.3.3

Combine the numerators over the common denominator.

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x^{\frac{2 - 1}{2}}} d x$

Step 7.2.3.4

Subtract $1$ from $2$ .

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 7.3

Apply basic rules of exponents.

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Step 7.3.1

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 7.3.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 7.3.2.1

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

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int x^{\frac{1}{2} \cdot - 1} d x$

Step 7.3.2.2

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

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int x^{\frac{- 1}{2}} d x$

Step 7.3.2.3

Move the negative in front of the fraction.

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + \int x^{- \frac{1}{2}} d x$

Step 8

By the Power Rule, the integral of $x^{- \frac{1}{2}}$ with respect to $x$ is $2 x^{\frac{1}{2}}$ .

$\sec (x) + C + 3 (\frac{x^{2}}{2} + C) + 2 x^{\frac{1}{2}} + C$

Step 9

Simplify.

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Step 9.1

Simplify.

$\sec (x) + \frac{3 x^{2}}{2} + 2 x^{\frac{1}{2}} + C$

Step 9.2

Reorder terms.

$\sec (x) + \frac{3}{2} x^{2} + 2 x^{\frac{1}{2}} + C$