Calculus Examples

$\int_{0}^{2} \frac{3 x^{2} - 1}{x^{2}} d x$

Step 1

This integral could not be completed using integration by parts. Mathway will use another method.

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\int_{0}^{2} (3 x^{2} - 1) {(x^{2})}^{- 1} d x$

Step 2.2

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

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Step 2.2.1

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

$\int_{0}^{2} (3 x^{2} - 1) x^{2 \cdot - 1} d x$

Step 2.2.2

Multiply $2$ by $- 1$ .

$\int_{0}^{2} (3 x^{2} - 1) x^{- 2} d x$

Step 3

Multiply $(3 x^{2} - 1) x^{- 2}$ .

$\int_{0}^{2} 3 x^{2} x^{- 2} - 1 x^{- 2} d x$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

Move $x^{- 2}$ .

$\int_{0}^{2} 3 (x^{- 2} x^{2}) - 1 x^{- 2} d x$

Step 4.1.2

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

$\int_{0}^{2} 3 x^{- 2 + 2} - 1 x^{- 2} d x$

Step 4.1.3

Add $- 2$ and $2$ .

$\int_{0}^{2} 3 x^{0} - 1 x^{- 2} d x$

Step 4.2

Simplify $3 x^{0}$ .

$\int_{0}^{2} 3 - 1 x^{- 2} d x$

Step 4.3

Rewrite $- 1 x^{- 2}$ as $- x^{- 2}$ .

$\int_{0}^{2} 3 - x^{- 2} d x$

Step 5

Split the single integral into multiple integrals.

$\int_{0}^{2} 3 d x + \int_{0}^{2} - x^{- 2} d x$

Step 6

Apply the constant rule.

$3 x]_{0}^{2} + \int_{0}^{2} - x^{- 2} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$3 x]_{0}^{2} - \int_{0}^{2} x^{- 2} d x$

Step 8

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

$3 x]_{0}^{2} - (- x^{- 1}]_{0}^{2})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $3 x$ at $2$ and at $0$ .

$(3 \cdot 2) - 3 \cdot 0 - (- x^{- 1}]_{0}^{2})$

Step 9.2

Evaluate $- x^{- 1}$ at $2$ and at $0$ .

$3 \cdot 2 - 3 \cdot 0 - (- 2^{- 1} + 0^{- 1})$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $3$ by $2$ .

$6 - 3 \cdot 0 - (- 2^{- 1} + 0^{- 1})$

Step 9.3.2

Multiply $- 3$ by $0$ .

$6 + 0 - (- 2^{- 1} + 0^{- 1})$

Step 9.3.3

Add $6$ and $0$ .

$6 - (- 2^{- 1} + 0^{- 1})$

Step 9.3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 - (- \frac{1}{2} + 0^{- 1})$

Step 9.3.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 - (- \frac{1}{2} + \frac{1}{0})$

Step 9.3.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$6 - (- \frac{1}{2} + \frac{1}{0})$

Step 9.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$6 - (- \frac{1}{2} + \frac{1}{0})$

Step 10

The expression contains a division by $0$ . The expression is undefined.

Undefined