Calculus Examples

$\int \frac{x^{2} + 3 x + 7}{\sqrt{x}} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

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

$\int \frac{x^{2} + 3 x + 7}{x^{\frac{1}{2}}} d x$

Step 3

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

$\int (x^{2} + 3 x + 7) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 4

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

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Step 4.1

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

$\int (x^{2} + 3 x + 7) x^{\frac{1}{2} \cdot - 1} d x$

Step 4.2

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

$\int (x^{2} + 3 x + 7) x^{\frac{- 1}{2}} d x$

Step 4.3

Move the negative in front of the fraction.

$\int (x^{2} + 3 x + 7) x^{- \frac{1}{2}} d x$

Step 5

Expand $(x^{2} + 3 x + 7) x^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\int (x^{2} + 3 x) x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.2

Apply the distributive property.

$\int x^{2} x^{- \frac{1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.3

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

$\int x^{2 - \frac{1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int x^{2 \cdot \frac{2}{2} - \frac{1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.5

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

$\int x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.6

Combine the numerators over the common denominator.

$\int x^{\frac{2 \cdot 2 - 1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $2$ by $2$ .

$\int x^{\frac{4 - 1}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.7.2

Subtract $1$ from $4$ .

$\int x^{\frac{3}{2}} + 3 x \cdot x^{- \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.8

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

$\int x^{\frac{3}{2}} + 3 (x^{1} x^{- \frac{1}{2}}) + 7 x^{- \frac{1}{2}} d x$

Step 5.9

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

$\int x^{\frac{3}{2}} + 3 x^{1 - \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.10

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

$\int x^{\frac{3}{2}} + 3 x^{\frac{2}{2} - \frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.11

Combine the numerators over the common denominator.

$\int x^{\frac{3}{2}} + 3 x^{\frac{2 - 1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.12

Subtract $1$ from $2$ .

$\int x^{\frac{3}{2}} + 3 x^{\frac{1}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.13

Reorder $x^{\frac{3}{2}}$ and $3 x^{\frac{1}{2}}$ .

$\int 3 x^{\frac{1}{2}} + x^{\frac{3}{2}} + 7 x^{- \frac{1}{2}} d x$

Step 5.14

Move $x^{\frac{3}{2}}$ .

$\int 3 x^{\frac{1}{2}} + 7 x^{- \frac{1}{2}} + x^{\frac{3}{2}} d x$

Step 5.15

Reorder $3 x^{\frac{1}{2}}$ and $7 x^{- \frac{1}{2}}$ .

$\int 7 x^{- \frac{1}{2}} + 3 x^{\frac{1}{2}} + x^{\frac{3}{2}} d x$

Step 6

Split the single integral into multiple integrals.

$\int 7 x^{- \frac{1}{2}} d x + \int 3 x^{\frac{1}{2}} d x + \int x^{\frac{3}{2}} d x$

Step 7

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

$7 \int x^{- \frac{1}{2}} d x + \int 3 x^{\frac{1}{2}} d x + \int x^{\frac{3}{2}} d x$

Step 8

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

$7 (2 x^{\frac{1}{2}} + C) + \int 3 x^{\frac{1}{2}} d x + \int x^{\frac{3}{2}} d x$

Step 9

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

$7 (2 x^{\frac{1}{2}} + C) + 3 \int x^{\frac{1}{2}} d x + \int x^{\frac{3}{2}} d x$

Step 10

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$7 (2 x^{\frac{1}{2}} + C) + 3 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int x^{\frac{3}{2}} d x$

Step 11

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$7 (2 x^{\frac{1}{2}} + C) + 3 (\frac{2}{3} x^{\frac{3}{2}} + C) + \frac{2}{5} x^{\frac{5}{2}} + C$

Step 12

Simplify.

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Step 12.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$7 (2 x^{\frac{1}{2}} + C) + 3 (\frac{2 x^{\frac{3}{2}}}{3} + C) + \frac{2}{5} x^{\frac{5}{2}} + C$

Step 12.2

Simplify.

$14 x^{\frac{1}{2}} + 2 x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C$