Calculus Examples

$\frac{1}{2} x^{2} \ln (x) - \frac{3}{4} x^{2}$

Step 1

Write $\frac{1}{2} x^{2} \ln (x) - \frac{3}{4} x^{2}$ as a function.

$f (x) = \frac{1}{2} x^{2} \ln (x) - \frac{3}{4} x^{2}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{1}{2} \cdot (x^{2} \ln (x)) - \frac{3}{4} x^{2} d x$

Step 4

Simplify.

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Step 4.1

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

$\int \frac{x^{2}}{2} \cdot \ln (x) - \frac{3}{4} x^{2} d x$

Step 4.2

Combine $\frac{x^{2}}{2}$ and $\ln (x)$ .

$\int \frac{x^{2} \ln (x)}{2} - \frac{3}{4} x^{2} d x$

Step 5

Split the single integral into multiple integrals.

$\int \frac{x^{2} \ln (x)}{2} d x + \int - \frac{3}{4} x^{2} d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int x^{2} \ln (x) d x + \int - \frac{3}{4} x^{2} d x$

Step 7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{2}$ .

$\frac{1}{2} (\ln (x) (\frac{1}{3} x^{3}) - \int \frac{1}{3} x^{3} \frac{1}{x} d x) + \int - \frac{3}{4} x^{2} d x$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{2} (\ln (x) \frac{x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x) + \int - \frac{3}{4} x^{2} d x$

Step 8.2

Combine $\ln (x)$ and $\frac{x^{3}}{3}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \int \frac{1}{3} x^{3} \frac{1}{x} d x) + \int - \frac{3}{4} x^{2} d x$

Step 8.3

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8.4

Multiply $\frac{x^{3}}{3}$ by $\frac{1}{x}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \int \frac{x^{3}}{3 x} d x) + \int - \frac{3}{4} x^{2} d x$

Step 8.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 8.5.1

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

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

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $x$ out of $3 x$ .

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

Step 8.5.2.2

Cancel the common factor.

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

Step 8.5.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \int \frac{x^{2}}{3} d x) + \int - \frac{3}{4} x^{2} d x$

Step 9

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - (\frac{1}{3} \int x^{2} d x)) + \int - \frac{3}{4} x^{2} d x$

Step 10

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

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{1}{3} x^{3} + C)) + \int - \frac{3}{4} x^{2} d x$

Step 11

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{x^{3}}{3} + C)) + \int - \frac{3}{4} x^{2} d x$

Step 12

Since $- \frac{3}{4}$ is constant with respect to $x$ , move $- \frac{3}{4}$ out of the integral.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{x^{3}}{3} + C)) - \frac{3}{4} \int x^{2} d x$

Step 13

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

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{x^{3}}{3} + C)) - \frac{3}{4} (\frac{1}{3} x^{3} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3}{4} (\frac{1}{3} x^{3}) + C$

Step 14.2

Simplify.

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Step 14.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3}{4} \cdot \frac{x^{3}}{3} + C$

Step 14.2.2

Multiply $\frac{x^{3}}{3}$ by $\frac{3}{4}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{x^{3} \cdot 3}{3 \cdot 4} + C$

Step 14.2.3

Multiply $3$ by $4$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{x^{3} \cdot 3}{12} + C$

Step 14.2.4

Move $3$ to the left of $x^{3}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3 \cdot x^{3}}{12} + C$

Step 14.2.5

Cancel the common factor of $3$ and $12$ .

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Step 14.2.5.1

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

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3 (x^{3})}{12} + C$

Step 14.2.5.2

Cancel the common factors.

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Step 14.2.5.2.1

Factor $3$ out of $12$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3 x^{3}}{3 \cdot 4} + C$

Step 14.2.5.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{3 x^{3}}{3 \cdot 4} + C$

Step 14.2.5.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - \frac{x^{3}}{4} + C$

Step 14.2.6

To write $\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) \cdot \frac{4}{4} - \frac{x^{3}}{4} + C$

Step 14.2.7

Combine $\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9})$ and $\frac{4}{4}$ .

$\frac{\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) \cdot 4}{4} - \frac{x^{3}}{4} + C$

Step 14.2.8

Combine the numerators over the common denominator.

$\frac{\frac{1}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) \cdot 4 - x^{3}}{4} + C$

Step 14.2.9

Combine $4$ and $\frac{1}{2}$ .

$\frac{\frac{4}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 14.2.10

Cancel the common factor of $4$ and $2$ .

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Step 14.2.10.1

Factor $2$ out of $4$ .

$\frac{\frac{2 \cdot 2}{2} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 14.2.10.2

Cancel the common factors.

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Step 14.2.10.2.1

Factor $2$ out of $2$ .

$\frac{\frac{2 \cdot 2}{2 (1)} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 14.2.10.2.2

Cancel the common factor.

$\frac{\frac{2 \cdot 2}{2 \cdot 1} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 14.2.10.2.3

Rewrite the expression.

$\frac{\frac{2}{1} (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 14.2.10.2.4

Divide $2$ by $1$ .

$\frac{2 (\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9}) - x^{3}}{4} + C$

Step 15

Reorder terms.

$\frac{1}{4} (2 (\frac{1}{3} \ln (x) x^{3} - \frac{1}{9} x^{3}) - x^{3}) + C$

Step 16

The answer is the antiderivative of the function $f (x) = \frac{1}{2} \cdot (x^{2} \ln (x)) - \frac{3}{4} x^{2}$ .

$F (x) =$ $\frac{1}{4} (2 (\frac{1}{3} \ln (x) x^{3} - \frac{1}{9} x^{3}) - x^{3}) + C$