Calculus Examples

$\sin (\ln (x))$

Step 1

Write $\sin (\ln (x))$ as a function.

$f (x) = \sin (\ln (x))$

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.

$F (x) = \int \sin (\ln (x)) d x$

Step 4

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

$\int e^{u} \sin (u) d u$

Step 5

Reorder $e^{u}$ and $\sin (u)$ .

$\int \sin (u) e^{u} d u$

Step 6

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = \sin (u)$ and $dv = e^{u}$ .

$\sin (u) e^{u} - \int e^{u} \cos (u) d u$

Step 7

Reorder $e^{u}$ and $\cos (u)$ .

$\sin (u) e^{u} - \int \cos (u) e^{u} d u$

Step 8

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = \cos (u)$ and $dv = e^{u}$ .

$\sin (u) e^{u} - (\cos (u) e^{u} - \int e^{u} (- \sin (u)) d u)$

Step 9

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\sin (u) e^{u} - (\cos (u) e^{u} - - \int e^{u} (\sin (u)) d u)$

Step 10

Simplify by multiplying through.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$\sin (u) e^{u} - (\cos (u) e^{u} + 1 \int e^{u} (\sin (u)) d u)$

Step 10.2

Multiply $\int e^{u} (\sin (u)) d u$ by $1$ .

$\sin (u) e^{u} - (\cos (u) e^{u} + \int e^{u} (\sin (u)) d u)$

Step 10.3

Apply the distributive property.

$\sin (u) e^{u} - (\cos (u) e^{u}) - \int e^{u} (\sin (u)) d u$

Step 11

Solving for $\int e^{u} \sin (u) d u$ , we find that $\int e^{u} \sin (u) d u$ = $\frac{\sin (u) e^{u} - (\cos (u) e^{u})}{2}$ .

$\frac{\sin (u) e^{u} - (\cos (u) e^{u})}{2} + C$

Step 12

Rewrite $\frac{\sin (u) e^{u} - \cos (u) e^{u}}{2} + C$ as $\frac{1}{2} (\sin (u) e^{u} - \cos (u) e^{u}) + C$ .

$\frac{1}{2} (\sin (u) e^{u} - \cos (u) e^{u}) + C$

Step 13

Replace all occurrences of $u$ with $\ln (x)$ .

$\frac{1}{2} (\sin (\ln (x)) e^{\ln (x)} - \cos (\ln (x)) e^{\ln (x)}) + C$

Step 14

Simplify.

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Step 14.1

Exponentiation and log are inverse functions.

$\frac{1}{2} (\sin (\ln (x)) x - \cos (\ln (x)) e^{\ln (x)}) + C$

Step 14.2

Exponentiation and log are inverse functions.

$\frac{1}{2} (\sin (\ln (x)) x - \cos (\ln (x)) x) + C$

Step 14.3

Apply the distributive property.

$\frac{1}{2} (\sin (\ln (x)) x) + \frac{1}{2} (- \cos (\ln (x)) x) + C$

Step 14.4

Multiply $\frac{1}{2} (\sin (\ln (x)) x)$ .

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Step 14.4.1

Combine $\sin (\ln (x))$ and $\frac{1}{2}$ .

$\frac{\sin (\ln (x))}{2} x + \frac{1}{2} (- \cos (\ln (x)) x) + C$

Step 14.4.2

Combine $\frac{\sin (\ln (x))}{2}$ and $x$ .

$\frac{\sin (\ln (x)) x}{2} + \frac{1}{2} (- \cos (\ln (x)) x) + C$

Step 14.5

Multiply $\frac{1}{2} (- \cos (\ln (x)) x)$ .

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Step 14.5.1

Combine $\frac{1}{2}$ and $\cos (\ln (x))$ .

$\frac{\sin (\ln (x)) x}{2} - \frac{\cos (\ln (x))}{2} x + C$

Step 14.5.2

Combine $x$ and $\frac{\cos (\ln (x))}{2}$ .

$\frac{\sin (\ln (x)) x}{2} - \frac{x \cos (\ln (x))}{2} + C$

Step 14.6

Reorder factors in $\frac{\sin (\ln (x)) x}{2} - \frac{x \cos (\ln (x))}{2}$ .

$\frac{x \sin (\ln (x))}{2} - \frac{x \cos (\ln (x))}{2} + C$

Step 15

Remove parentheses.

$\frac{1}{2} x \sin (\ln (x)) - \frac{1}{2} x \cos (\ln (x)) + C$

Step 16

The answer is the antiderivative of the function $f (x) = \sin (\ln (x))$ .

$F (x) =$ $\frac{1}{2} x \sin (\ln (x)) - \frac{1}{2} x \cos (\ln (x)) + C$