Calculus Examples

$f (x) = 3 x^{2} \sin (x^{3} + 2003)$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 3 x^{2} \sin (x^{3} + 2003) d x$

Step 3

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

$3 \int x^{2} \sin (x^{3} + 2003) d x$

Step 4

Let $u = x^{3} + 2003$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x^{3} + 2003$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x^{3} + 2003$ .

$\frac{d}{d x} [x^{3} + 2003]$

Step 4.1.2

By the Sum Rule, the derivative of $x^{3} + 2003$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2003]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2003]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [2003]$

Step 4.1.4

Since $2003$ is constant with respect to $x$ , the derivative of $2003$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 4.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 4.2

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

$3 \int \sin (u) \frac{1}{3} d u$

Step 5

Combine $\sin (u)$ and $\frac{1}{3}$ .

$3 \int \frac{\sin (u)}{3} d u$

Step 6

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$3 (\frac{1}{3} \int \sin (u) d u)$

Step 7

Simplify.

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Step 7.1

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

$\frac{3}{3} \int \sin (u) d u$

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Cancel the common factor.

$\frac{3}{3} \int \sin (u) d u$

Step 7.2.2

Rewrite the expression.

$1 \int \sin (u) d u$

Step 7.3

Multiply $\int \sin (u) d u$ by $1$ .

$\int \sin (u) d u$

Step 8

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$- \cos (u) + C$

Step 9

Replace all occurrences of $u$ with $x^{3} + 2003$ .

$- \cos (x^{3} + 2003) + C$

Step 10

The answer is the antiderivative of the function $f (x) = 3 x^{2} \sin (x^{3} + 2003)$ .

$F (x) =$ $- \cos (x^{3} + 2003) + C$