Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = {(2 x + 3)}^{2}$ and $d v = e^{x + 1}$ .

${(2 x + 3)}^{2} e^{x + 1} - \int e^{x + 1} (8 x + 12) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 8 x + 12$ and $d v = e^{x + 1}$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - \int e^{x + 1} \cdot 8 d x)$

Step 3

Move $8$ to the left of $e^{x + 1}$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - \int 8 e^{x + 1} d x)$

Step 4

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

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - (8 \int e^{x + 1} d x))$

Step 5

Multiply $8$ by $- 1$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 \int e^{x + 1} d x)$

Step 6

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 6.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 6.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 \int e^{u} d u)$

Step 7

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 (e^{u} + C))$

Step 8

Simplify.

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Step 8.1

Rewrite ${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 (e^{u} + C))$ as $(2 x + 3) (2 x + 3) e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$ .

$(2 x + 3) (2 x + 3) e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Raise $2 x + 3$ to the power of $1$ .

${(2 x + 3)}^{1} (2 x + 3) e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$

Step 8.2.2

Raise $2 x + 3$ to the power of $1$ .

${(2 x + 3)}^{1} {(2 x + 3)}^{1} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$

Step 8.2.3

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

${(2 x + 3)}^{1 + 1} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$

Step 8.2.4

Add $1$ and $1$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{u}) + C$

Step 9

Replace all occurrences of $u$ with $x + 1$ .

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1} - 8 e^{x + 1}) + C$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

${(2 x + 3)}^{2} e^{x + 1} - ((8 x + 12) e^{x + 1}) - (- 8 e^{x + 1}) + C$

Step 10.2

Multiply $- 8$ by $- 1$ .

${(2 x + 3)}^{2} e^{x + 1} - (8 x + 12) e^{x + 1} + 8 e^{x + 1} + C$

Step 10.3

Reorder factors in $- (8 x + 12) e^{x + 1} + 8 e^{x + 1}$ .

${(2 x + 3)}^{2} e^{x + 1} - e^{x + 1} (8 x + 12) + 8 e^{x + 1} + C$