Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify.

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Step 2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2

Use the Binomial Theorem.

$\int 3 (1^{4} + 4 \cdot 1^{3} (2 x) + 6 \cdot 1^{2} {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3

Simplify each term.

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Step 2.3.1

One to any power is one.

$\int 3 (1 + 4 \cdot 1^{3} (2 x) + 6 \cdot 1^{2} {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.2

One to any power is one.

$\int 3 (1 + 4 \cdot 1 (2 x) + 6 \cdot 1^{2} {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.3

Multiply $4$ by $1$ .

$\int 3 (1 + 4 (2 x) + 6 \cdot 1^{2} {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.4

Multiply $2$ by $4$ .

$\int 3 (1 + 8 x + 6 \cdot 1^{2} {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.5

One to any power is one.

$\int 3 (1 + 8 x + 6 \cdot 1 {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.6

Multiply $6$ by $1$ .

$\int 3 (1 + 8 x + 6 {(2 x)}^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.7

Apply the product rule to $2 x$ .

$\int 3 (1 + 8 x + 6 (2^{2} x^{2}) + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.8

Raise $2$ to the power of $2$ .

$\int 3 (1 + 8 x + 6 (4 x^{2}) + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.9

Multiply $4$ by $6$ .

$\int 3 (1 + 8 x + 24 x^{2} + 4 \cdot 1 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.10

Multiply $4$ by $1$ .

$\int 3 (1 + 8 x + 24 x^{2} + 4 {(2 x)}^{3} + {(2 x)}^{4}) x d x$

Step 2.3.11

Apply the product rule to $2 x$ .

$\int 3 (1 + 8 x + 24 x^{2} + 4 (2^{3} x^{3}) + {(2 x)}^{4}) x d x$

Step 2.3.12

Raise $2$ to the power of $3$ .

$\int 3 (1 + 8 x + 24 x^{2} + 4 (8 x^{3}) + {(2 x)}^{4}) x d x$

Step 2.3.13

Multiply $8$ by $4$ .

$\int 3 (1 + 8 x + 24 x^{2} + 32 x^{3} + {(2 x)}^{4}) x d x$

Step 2.3.14

Apply the product rule to $2 x$ .

$\int 3 (1 + 8 x + 24 x^{2} + 32 x^{3} + 2^{4} x^{4}) x d x$

Step 2.3.15

Raise $2$ to the power of $4$ .

$\int 3 (1 + 8 x + 24 x^{2} + 32 x^{3} + 16 x^{4}) x d x$

Step 2.4

Apply the distributive property.

$\int (3 \cdot 1 + 3 (8 x) + 3 (24 x^{2}) + 3 (32 x^{3}) + 3 (16 x^{4})) x d x$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $3$ by $1$ .

$\int (3 + 3 (8 x) + 3 (24 x^{2}) + 3 (32 x^{3}) + 3 (16 x^{4})) x d x$

Step 2.5.2

Multiply $8$ by $3$ .

$\int (3 + 24 x + 3 (24 x^{2}) + 3 (32 x^{3}) + 3 (16 x^{4})) x d x$

Step 2.5.3

Multiply $24$ by $3$ .

$\int (3 + 24 x + 72 x^{2} + 3 (32 x^{3}) + 3 (16 x^{4})) x d x$

Step 2.5.4

Multiply $32$ by $3$ .

$\int (3 + 24 x + 72 x^{2} + 96 x^{3} + 3 (16 x^{4})) x d x$

Step 2.5.5

Multiply $16$ by $3$ .

$\int (3 + 24 x + 72 x^{2} + 96 x^{3} + 48 x^{4}) x d x$

Step 2.6

Apply the distributive property.

$\int 3 x + 24 x \cdot x + 72 x^{2} x + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7

Simplify.

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Step 2.7.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.7.1.1

Move $x$ .

$\int 3 x + 24 (x \cdot x) + 72 x^{2} x + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.1.2

Multiply $x$ by $x$ .

$\int 3 x + 24 x^{2} + 72 x^{2} x + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.2

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

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Step 2.7.2.1

Move $x$ .

$\int 3 x + 24 x^{2} + 72 (x \cdot x^{2}) + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.2.2

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

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Step 2.7.2.2.1

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

$\int 3 x + 24 x^{2} + 72 (x^{1} x^{2}) + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.2.2.2

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

$\int 3 x + 24 x^{2} + 72 x^{1 + 2} + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.2.3

Add $1$ and $2$ .

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{3} x + 48 x^{4} x d x$

Step 2.7.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.7.3.1

Move $x$ .

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 (x \cdot x^{3}) + 48 x^{4} x d x$

Step 2.7.3.2

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

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Step 2.7.3.2.1

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

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 (x^{1} x^{3}) + 48 x^{4} x d x$

Step 2.7.3.2.2

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

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{1 + 3} + 48 x^{4} x d x$

Step 2.7.3.3

Add $1$ and $3$ .

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{4} + 48 x^{4} x d x$

Step 2.7.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.7.4.1

Move $x$ .

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{4} + 48 (x \cdot x^{4}) d x$

Step 2.7.4.2

Multiply $x$ by $x^{4}$ .

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Step 2.7.4.2.1

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

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{4} + 48 (x^{1} x^{4}) d x$

Step 2.7.4.2.2

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

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{4} + 48 x^{1 + 4} d x$

Step 2.7.4.3

Add $1$ and $4$ .

$\int 3 x + 24 x^{2} + 72 x^{3} + 96 x^{4} + 48 x^{5} d x$

Step 3

Split the single integral into multiple integrals.

$\int 3 x d x + \int 24 x^{2} d x + \int 72 x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 4

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

$3 \int x d x + \int 24 x^{2} d x + \int 72 x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 5

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

$3 (\frac{1}{2} x^{2} + C) + \int 24 x^{2} d x + \int 72 x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 6

Since $24$ is constant with respect to $x$ , move $24$ out of the integral.

$3 (\frac{1}{2} x^{2} + C) + 24 \int x^{2} d x + \int 72 x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 7

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

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + \int 72 x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 8

Since $72$ is constant with respect to $x$ , move $72$ out of the integral.

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 \int x^{3} d x + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 9

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

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 (\frac{1}{4} x^{4} + C) + \int 96 x^{4} d x + \int 48 x^{5} d x$

Step 10

Since $96$ is constant with respect to $x$ , move $96$ out of the integral.

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 (\frac{1}{4} x^{4} + C) + 96 \int x^{4} d x + \int 48 x^{5} d x$

Step 11

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 (\frac{1}{4} x^{4} + C) + 96 (\frac{1}{5} x^{5} + C) + \int 48 x^{5} d x$

Step 12

Since $48$ is constant with respect to $x$ , move $48$ out of the integral.

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 (\frac{1}{4} x^{4} + C) + 96 (\frac{1}{5} x^{5} + C) + 48 \int x^{5} d x$

Step 13

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

$3 (\frac{1}{2} x^{2} + C) + 24 (\frac{1}{3} x^{3} + C) + 72 (\frac{1}{4} x^{4} + C) + 96 (\frac{1}{5} x^{5} + C) + 48 (\frac{1}{6} x^{6} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + 48 (\frac{1}{6} x^{6}) + C$

Step 14.2

Simplify.

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Step 14.2.1

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

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + 48 \frac{x^{6}}{6} + C$

Step 14.2.2

Combine $48$ and $\frac{x^{6}}{6}$ .

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + \frac{48 x^{6}}{6} + C$

Step 14.2.3

Cancel the common factor of $48$ and $6$ .

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Step 14.2.3.1

Factor $6$ out of $48 x^{6}$ .

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + \frac{6 (8 x^{6})}{6} + C$

Step 14.2.3.2

Cancel the common factors.

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Step 14.2.3.2.1

Factor $6$ out of $6$ .

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + \frac{6 (8 x^{6})}{6 (1)} + C$

Step 14.2.3.2.2

Cancel the common factor.

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + \frac{6 (8 x^{6})}{6 \cdot 1} + C$

Step 14.2.3.2.3

Rewrite the expression.

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + \frac{8 x^{6}}{1} + C$

Step 14.2.3.2.4

Divide $8 x^{6}$ by $1$ .

$\frac{3 x^{2}}{2} + 8 x^{3} + 18 x^{4} + \frac{96 x^{5}}{5} + 8 x^{6} + C$

Step 14.3

Reorder terms.

$\frac{3}{2} x^{2} + 8 x^{3} + 18 x^{4} + \frac{96}{5} x^{5} + 8 x^{6} + C$