Calculus Examples

$\int (2 + 3 x) x d x$

Step 1

Multiply $(2 + 3 x) x$ .

$\int 2 x + 3 x \cdot x d x$

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Cancel the common factor.

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

Step 8.2.2.2

Rewrite the expression.

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

Step 8.2.3

Multiply $x^{2}$ by $1$ .

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

Step 8.2.4

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

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

Step 8.2.5

Cancel the common factor of $3$ .

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Step 8.2.5.1

Cancel the common factor.

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

Step 8.2.5.2

Rewrite the expression.

$x^{2} + 1 x^{3} + C$

Step 8.2.6

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

$x^{2} + x^{3} + C$