Calculus Examples

\int_{2 x}^{3 x + 1} sin (t^{4}) d t

$\int_{2 x}^{3 x + 1} \sin (t^{4}) d t$

Step 1

Split the integral into two integrals where

c

$c$ is some value between

2 x

$2 x$ and

3 x + 1

$3 x + 1$ .

\frac{d}{d x} [\int_{2 x}^{c} sin (t^{4}) d t + \int_{c}^{3 x + 1} sin (t^{4}) d t]

$\frac{d}{d x} [\int_{2 x}^{c} \sin (t^{4}) d t + \int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 2

By the Sum Rule, the derivative of

\int_{2 x}^{c} sin (t^{4}) d t + \int_{c}^{3 x + 1} sin (t^{4}) d t

$\int_{2 x}^{c} \sin (t^{4}) d t + \int_{c}^{3 x + 1} \sin (t^{4}) d t$ with respect to

x

$x$ is

\frac{d}{d x} [\int_{2 x}^{c} sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$\frac{d}{d x} [\int_{2 x}^{c} \sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$ .

\frac{d}{d x} [\int_{2 x}^{c} sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$\frac{d}{d x} [\int_{2 x}^{c} \sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 3

Swap the bounds of integration.

\frac{d}{d x} [- \int_{c}^{2 x} sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$\frac{d}{d x} [- \int_{c}^{2 x} \sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 4

Take the derivative of

- \int_{c}^{2 x} sin (t^{4}) d t

$- \int_{c}^{2 x} \sin (t^{4}) d t$ with respect to

x

$x$ using Fundamental Theorem of Calculus and the chain rule.

\frac{d}{d x} [2 x] (- sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$\frac{d}{d x} [2 x] (- \sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 5

Differentiate.

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Step 5.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x

$2 x$ with respect to

x

$x$ is

2 \frac{d}{d x} [x]

$2 \frac{d}{d x} [x]$ .

2 \frac{d}{d x} [x] (- sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$2 \frac{d}{d x} [x] (- \sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 5.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

2 \cdot 1 (- sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$2 \cdot 1 (- \sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 5.3

Multiply

2

$2$ by

1

$1$ .

2 (- sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} sin (t^{4}) d t]

$2 (- \sin ({(2 x)}^{4})) + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$

Step 6

Take the derivative of

$\int_{c}^{3 x + 1} \sin (t^{4}) d t$ with respect to

$x$ using Fundamental Theorem of Calculus and the chain rule.

$2 (- \sin ({(2 x)}^{4})) + \frac{d}{d x} [3 x + 1] \sin ({(3 x + 1)}^{4})$

Step 7

By the Sum Rule, the derivative of

$3 x + 1$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$2 (- \sin ({(2 x)}^{4})) + (\frac{d}{d x} [3 x] + \frac{d}{d x} [1]) \sin ({(3 x + 1)}^{4})$

Step 8

Evaluate

$\frac{d}{d x} [3 x]$ .

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Step 8.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

$2 (- \sin ({(2 x)}^{4})) + (3 \frac{d}{d x} [x] + \frac{d}{d x} [1]) \sin ({(3 x + 1)}^{4})$

Step 8.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 (- \sin ({(2 x)}^{4})) + (3 \cdot 1 + \frac{d}{d x} [1]) \sin ({(3 x + 1)}^{4})$

Step 8.3

Multiply

$3$ by

$1$ .

$2 (- \sin ({(2 x)}^{4})) + (3 + \frac{d}{d x} [1]) \sin ({(3 x + 1)}^{4})$

Step 9

Differentiate using the Constant Rule.

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Step 9.1

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$2 (- \sin ({(2 x)}^{4})) + (3 + 0) \sin ({(3 x + 1)}^{4})$

Step 9.2

Simplify with factoring out.

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Step 9.2.1

Add

$3$ and

$0$ .

$2 (- \sin ({(2 x)}^{4})) + 3 \sin ({(3 x + 1)}^{4})$

Step 9.2.2

Factor

$2$ out of

$2 x$ .

$2 (- \sin ({(2 (x))}^{4})) + 3 \sin ({(3 x + 1)}^{4})$

Step 9.2.3

Simplify the expression.

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Step 9.2.3.1

Apply the product rule to

$2 (x)$ .

$2 (- \sin (2^{4} x^{4})) + 3 \sin ({(3 x + 1)}^{4})$

Step 9.2.3.2

Raise

$2$ to the power of

$4$ .

$2 (- \sin (16 x^{4})) + 3 \sin ({(3 x + 1)}^{4})$

Step 9.2.3.3

Multiply

$- 1$ by

$2$ .

$- 2 \sin (16 x^{4}) + 3 \sin ({(3 x + 1)}^{4})$