Calculus Examples

$3 x^{2} + 8 x + 5 (d x)$

Step 1

Split the single integral into multiple integrals.

$\int 3 x^{2} d x + \int 8 x d x + \int 5 d x d x$

Step 2

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

$3 \int x^{2} d x + \int 8 x d x + \int 5 d x d x$

Step 3

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

$3 (\frac{1}{3} x^{3} + C) + \int 8 x d x + \int 5 d x d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$3 (\frac{1}{3} x^{3} + C) + 8 \int x d x + \int 5 d x d x$

Step 5

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

$3 (\frac{1}{3} x^{3} + C) + 8 (\frac{1}{2} x^{2} + C) + \int 5 d x d x$

Step 6

Since $5 d$ is constant with respect to $x$ , move $5 d$ out of the integral.

$3 (\frac{1}{3} x^{3} + C) + 8 (\frac{1}{2} x^{2} + C) + 5 d \int x d x$

Step 7

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

$3 (\frac{1}{3} x^{3} + C) + 8 (\frac{1}{2} x^{2} + C) + 5 d (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$x^{3} + 4 x^{2} + 5 d (\frac{1}{2} x^{2}) + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$x^{3} + 4 x^{2} + 5 d \frac{x^{2}}{2} + C$

Step 8.2.2

Combine $5$ and $\frac{x^{2}}{2}$ .

$x^{3} + 4 x^{2} + d \frac{5 x^{2}}{2} + C$

Step 8.2.3

Combine $d$ and $\frac{5 x^{2}}{2}$ .

$x^{3} + 4 x^{2} + \frac{d (5 x^{2})}{2} + C$

Step 8.2.4

Move $5$ to the left of $d$ .

$x^{3} + 4 x^{2} + \frac{5 d x^{2}}{2} + C$

$x^{3} + 4 x^{2} + \frac{5}{2} d x^{2} + C$