Calculus Examples

$\frac{d y}{d x} = e^{4 x} \cos (e^{4 x})$ , $y (0) = 0$

Step 1

Rewrite the equation.

$d y = e^{4 x} \cos (e^{4 x}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int e^{4 x} \cos (e^{4 x}) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int e^{4 x} \cos (e^{4 x}) d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.3.1.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 2.3.1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 2.3.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 2.3.1.1.3

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

$4 \cdot 1$

Step 2.3.1.1.4

Multiply $4$ by $1$ .

$4$

Step 2.3.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$y + C_{1} = \int e^{u_{1}} \cos (e^{u_{1}}) \frac{1}{4} d u_{1}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Combine $\frac{1}{4}$ and $e^{u_{1}}$ .

$y + C_{1} = \int \frac{e^{u_{1}}}{4} \cos (e^{u_{1}}) d u_{1}$

Step 2.3.2.2

Combine $\frac{e^{u_{1}}}{4}$ and $\cos (e^{u_{1}})$ .

$y + C_{1} = \int \frac{e^{u_{1}} \cos (e^{u_{1}})}{4} d u_{1}$

Step 2.3.3

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$y + C_{1} = \frac{1}{4} \int e^{u_{1}} \cos (e^{u_{1}}) d u_{1}$

Step 2.3.4

Let $u_{2} = e^{u_{1}}$ . Then $d u_{2} = e^{u_{1}} d u_{1}$ , so $\frac{1}{e^{u_{1}}} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.3.4.1

Let $u_{2} = e^{u_{1}}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 2.3.4.1.1

Differentiate $e^{u_{1}}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}]$

Step 2.3.4.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}}$

Step 2.3.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{1} = \frac{1}{4} \int \cos (u_{2}) d u_{2}$

Step 2.3.5

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$y + C_{1} = \frac{1}{4} (\sin (u_{2}) + C_{2})$

Step 2.3.6

Simplify.

$y + C_{1} = \frac{1}{4} \sin (u_{2}) + C_{2}$

Step 2.3.7

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 2.3.7.1

Replace all occurrences of $u_{2}$ with $e^{u_{1}}$ .

$y + C_{1} = \frac{1}{4} \sin (e^{u_{1}}) + C_{2}$

Step 2.3.7.2

Replace all occurrences of $u_{1}$ with $4 x$ .

$y + C_{1} = \frac{1}{4} \sin (e^{4 x}) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = \frac{1}{4} \sin (e^{4 x}) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \frac{1}{4} \sin (e^{4 x}) + K$ .

$0 = \frac{1}{4} \sin (e^{4 \cdot 0}) + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{1}{4} \cdot \sin (e^{4 \cdot 0}) + K = 0$ .

$\frac{1}{4} \cdot \sin (e^{4 \cdot 0}) + K = 0$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $4$ by $0$ .

$\frac{1}{4} \cdot \sin (e^{0}) + K = 0$

Step 4.2.1.2

Anything raised to $0$ is $1$ .

$\frac{1}{4} \cdot \sin (1) + K = 0$

Step 4.2.1.3

Evaluate $\sin (1)$ .

$\frac{1}{4} \cdot 0.0174524 + K = 0$

Step 4.2.1.4

Combine $\frac{1}{4}$ and $0.0174524$ .

$\frac{0.0174524}{4} + K = 0$

Step 4.2.1.5

Divide $0.0174524$ by $4$ .

$0.0043631 + K = 0$

Step 4.3

Subtract $0.0043631$ from both sides of the equation.

$K = - 0.0043631$

Step 5

Substitute $- 0.0043631$ for $K$ in $y = \frac{1}{4} \sin (e^{4 x}) + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $- 0.0043631$ for $K$ .

$y = \frac{1}{4} \sin (e^{4 x}) - 0.0043631$

Step 5.2

Combine $\frac{1}{4}$ and $\sin (e^{4 x})$ .

$y = \frac{\sin (e^{4 x})}{4} - 0.0043631$