Calculus Examples

$\sin (2 x) \cdot e^{\sin^{2} (x)}$

Step 1

Write $\sin (2 x) \cdot e^{\sin^{2} (x)}$ as a function.

$f (x) = \sin (2 x) \cdot e^{\sin^{2} (x)}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int \sin (2 x) \cdot e^{\sin^{2} (x)} d x$

Step 4

Let $u_{2} = \sin^{2} (x)$ . Then $d u_{2} = \sin (2 x) d x$ , so $\frac{1}{\sin (2 x)} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 4.1

Let $u_{2} = \sin^{2} (x)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $\sin^{2} (x)$ .

$\frac{d}{d x} [\sin^{2} (x)]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

Tap for more steps...

Step 4.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (x)]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\sin (x)]$

Step 4.1.2.3

Replace all occurrences of $u_{1}$ with $\sin (x)$ .

$2 \sin (x) \frac{d}{d x} [\sin (x)]$

Step 4.1.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$2 \sin (x) \cos (x)$

Step 4.1.4

Simplify.

Tap for more steps...

Step 4.1.4.1

Reorder the factors of $2 \sin (x) \cos (x)$ .

$2 \cos (x) \sin (x)$

Step 4.1.4.2

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x))$

Step 4.1.4.3

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x)$

Step 4.1.4.4

Apply the sine double-angle identity.

$\sin (2 x)$

Step 4.2

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

$\int e^{u_{2}} d u_{2}$

Step 5

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$e^{u_{2}} + C$

Step 6

Replace all occurrences of $u_{2}$ with $\sin^{2} (x)$ .

$e^{\sin^{2} (x)} + C$

Step 7

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

$F (x) =$ $e^{\sin^{2} (x)} + C$