Calculus Examples

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

Step 1

Remove parentheses.

$\int \sqrt{x} \sqrt{x} + \frac{3}{x^{2}} d x$

Step 2

Simplify.

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Step 2.1

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

$\int {\sqrt{x}}^{1} \sqrt{x} + \frac{3}{x^{2}} d x$

Step 2.2

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

$\int {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \frac{3}{x^{2}} d x$

Step 2.3

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

$\int {\sqrt{x}}^{1 + 1} + \frac{3}{x^{2}} d x$

Step 2.4

Add $1$ and $1$ .

$\int {\sqrt{x}}^{2} + \frac{3}{x^{2}} d x$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

$\int x^{\frac{1}{2} \cdot 2} + \frac{3}{x^{2}} d x$

Step 2.5.3

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

$\int x^{\frac{2}{2}} + \frac{3}{x^{2}} d x$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\int x^{\frac{2}{2}} + \frac{3}{x^{2}} d x$

Step 2.5.4.2

Rewrite the expression.

$\int x^{1} + \frac{3}{x^{2}} d x$

Step 2.5.5

Simplify.

$\int x + \frac{3}{x^{2}} d x$

Step 3

Split the single integral into multiple integrals.

$\int x d x + \int \frac{3}{x^{2}} d x$

Step 4

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

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

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Multiply $- 1$ by $3$ .

$\frac{1}{2} x^{2} - 3 x^{- 1} + C$