Calculus Examples

$\int (7 x^{2} - 5) (3 x^{3} + 4) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 7 x^{2} (3 x^{3} + 4) - 5 (3 x^{3} + 4) d x$

Step 1.2

Apply the distributive property.

$\int 7 x^{2} (3 x^{3}) + 7 x^{2} \cdot 4 - 5 (3 x^{3} + 4) d x$

Step 1.3

Apply the distributive property.

$\int 7 x^{2} (3 x^{3}) + 7 x^{2} \cdot 4 - 5 (3 x^{3}) - 5 \cdot 4 d x$

Step 1.4

Move $x^{2}$ .

$\int 7 \cdot 3 x^{2} x^{3} + 7 x^{2} \cdot 4 - 5 (3 x^{3}) - 5 \cdot 4 d x$

Step 1.5

Move $x^{2}$ .

$\int 7 \cdot 3 x^{2} x^{3} + 7 \cdot 4 x^{2} - 5 (3 x^{3}) - 5 \cdot 4 d x$

Step 1.6

Multiply $7$ by $3$ .

$\int 21 x^{2} x^{3} + 7 \cdot 4 x^{2} - 5 \cdot 3 x^{3} - 5 \cdot 4 d x$

Step 1.7

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

$\int 21 x^{2 + 3} + 7 \cdot 4 x^{2} - 5 \cdot 3 x^{3} - 5 \cdot 4 d x$

Step 1.8

Add $2$ and $3$ .

$\int 21 x^{5} + 7 \cdot 4 x^{2} - 5 \cdot 3 x^{3} - 5 \cdot 4 d x$

Step 1.9

Multiply $7$ by $4$ .

$\int 21 x^{5} + 28 x^{2} - 5 \cdot 3 x^{3} - 5 \cdot 4 d x$

Step 1.10

Multiply $- 5$ by $3$ .

$\int 21 x^{5} + 28 x^{2} - 15 x^{3} - 5 \cdot 4 d x$

Step 1.11

Multiply $- 5$ by $4$ .

$\int 21 x^{5} + 28 x^{2} - 15 x^{3} - 20 d x$

Step 1.12

Move $28 x^{2}$ .

$\int 21 x^{5} - 15 x^{3} + 28 x^{2} - 20 d x$

Step 2

Split the single integral into multiple integrals.

$\int 21 x^{5} d x + \int - 15 x^{3} d x + \int 28 x^{2} d x + \int - 20 d x$

Step 3

Since $21$ is constant with respect to $x$ , move $21$ out of the integral.

$21 \int x^{5} d x + \int - 15 x^{3} d x + \int 28 x^{2} d x + \int - 20 d x$

Step 4

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$21 (\frac{1}{6} x^{6} + C) + \int - 15 x^{3} d x + \int 28 x^{2} d x + \int - 20 d x$

Step 5

Since $- 15$ is constant with respect to $x$ , move $- 15$ out of the integral.

$21 (\frac{1}{6} x^{6} + C) - 15 \int x^{3} d x + \int 28 x^{2} d x + \int - 20 d x$

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$21 (\frac{1}{6} x^{6} + C) - 15 (\frac{1}{4} x^{4} + C) + \int 28 x^{2} d x + \int - 20 d x$

Step 7

Since $28$ is constant with respect to $x$ , move $28$ out of the integral.

$21 (\frac{1}{6} x^{6} + C) - 15 (\frac{1}{4} x^{4} + C) + 28 \int x^{2} d x + \int - 20 d x$

Step 8

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

$21 (\frac{1}{6} x^{6} + C) - 15 (\frac{1}{4} x^{4} + C) + 28 (\frac{1}{3} x^{3} + C) + \int - 20 d x$

Step 9

Apply the constant rule.

$21 (\frac{1}{6} x^{6} + C) - 15 (\frac{1}{4} x^{4} + C) + 28 (\frac{1}{3} x^{3} + C) - 20 x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$21 (\frac{x^{6}}{6} + C) - 15 (\frac{1}{4} x^{4} + C) + 28 (\frac{1}{3} x^{3} + C) - 20 x + C$

Step 10.1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$21 (\frac{x^{6}}{6} + C) - 15 (\frac{x^{4}}{4} + C) + 28 (\frac{1}{3} x^{3} + C) - 20 x + C$

Step 10.1.3

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

$21 (\frac{x^{6}}{6} + C) - 15 (\frac{x^{4}}{4} + C) + 28 (\frac{x^{3}}{3} + C) - 20 x + C$

Step 10.2

Simplify.

$\frac{7 x^{6}}{2} - \frac{15 x^{4}}{4} + \frac{28 x^{3}}{3} - 20 x + C$

Step 11

Reorder terms.

$\frac{7}{2} x^{6} - \frac{15}{4} x^{4} + \frac{28}{3} x^{3} - 20 x + C$