Calculus Examples

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

Step 1

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

Step 2

Remove parentheses.

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

Step 3

Simplify.

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Step 3.1

Factor $3$ out of $3 x$ .

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

Step 3.2

Apply the product rule to $3 (x)$ .

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

Step 3.3

Raise $3$ to the power of $2$ .

$\int x^{2} + \frac{1}{9 x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{2} d x + \int \frac{1}{9 x^{2}} d x$

Step 5

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

$\frac{1}{3} x^{3} + C + \int \frac{1}{9 x^{2}} d x$

Step 6

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{3} x^{3} + C + \frac{1}{9} \int \frac{1}{x^{2}} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

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

$\frac{1}{3} x^{3} + C + \frac{1}{9} \int {(x^{2})}^{- 1} d x$

Step 7.2

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

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Step 7.2.1

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

$\frac{1}{3} x^{3} + C + \frac{1}{9} \int x^{2 \cdot - 1} d x$

Step 7.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} x^{3} + C + \frac{1}{9} \int x^{- 2} d x$

Step 8

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

$\frac{1}{3} x^{3} + C + \frac{1}{9} (- x^{- 1} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{1}{3} x^{3} + \frac{1}{9} (- x^{- 1}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{1}{9}$ and $x^{- 1}$ .

$\frac{1}{3} x^{3} - \frac{x^{- 1}}{9} + C$

Step 9.2.2

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

$\frac{1}{3} x^{3} - \frac{1}{9 x} + C$