Calculus Examples

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

Step 1

Split the integral into two integrals where $c$ is some value between $2 x$ and $3 x + 1$ .

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

Step 2

By the Sum Rule, the derivative of $\int_{2 x}^{c} \sin (t^{4}) d t + \int_{c}^{3 x + 1} \sin (t^{4}) d t$ with respect to $x$ is $\frac{d}{d x} [\int_{2 x}^{c} \sin (t^{4}) d t] + \frac{d}{d x} [\int_{c}^{3 x + 1} \sin (t^{4}) d t]$ .

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

Step 3

Swap the bounds of integration.

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

Step 4

Take the derivative of $- \int_{c}^{2 x} \sin (t^{4}) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 5.2

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

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

Step 5.3

Multiply $2$ by $1$ .

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

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

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