Calculus Examples

$\int \frac{3 \sin (2 x)}{4 \sin^{2} (x) - 4} d x$

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $4 \sin^{2} (x)$ and $- 4$ .

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

Step 1.2

Factor $- 4$ out of $- 4$ .

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

Step 1.3

Factor $- 4$ out of $4 \sin^{2} (x)$ .

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

Step 1.4

Factor $- 4$ out of $- 4 (1) - 4 (- \sin^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

$\int \frac{3 \sin (2 x)}{- 4 \cos^{2} (x)} d x$

Step 3

Simplify the numerator.

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Step 3.1

Apply the sine double-angle identity.

$\int \frac{3 \cdot 2 \sin (x) \cos (x)}{- 4 \cos^{2} (x)} d x$

Step 3.2

Multiply $3$ by $2$ .

$\int \frac{6 \sin (x) \cos (x)}{- 4 \cos^{2} (x)} d x$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $6$ and $- 4$ .

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Step 4.1.1

Factor $2$ out of $6 \sin (x) \cos (x)$ .

$\int \frac{2 (3 \sin (x) \cos (x))}{- 4 \cos^{2} (x)} d x$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $- 4 \cos^{2} (x)$ .

$\int \frac{2 (3 \sin (x) \cos (x))}{2 (- 2 \cos^{2} (x))} d x$

Step 4.1.2.2

Cancel the common factor.

$\int \frac{2 (3 \sin (x) \cos (x))}{2 (- 2 \cos^{2} (x))} d x$

Step 4.1.2.3

Rewrite the expression.

$\int \frac{3 \sin (x) \cos (x)}{- 2 \cos^{2} (x)} d x$

Step 4.2

Cancel the common factor of $\cos (x)$ and $\cos^{2} (x)$ .

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Step 4.2.1

Factor $\cos (x)$ out of $3 \sin (x) \cos (x)$ .

$\int \frac{\cos (x) (3 \sin (x))}{- 2 \cos^{2} (x)} d x$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $\cos (x)$ out of $- 2 \cos^{2} (x)$ .

$\int \frac{\cos (x) (3 \sin (x))}{\cos (x) (- 2 \cos (x))} d x$

Step 4.2.2.2

Cancel the common factor.

$\int \frac{\cos (x) (3 \sin (x))}{\cos (x) (- 2 \cos (x))} d x$

Step 4.2.2.3

Rewrite the expression.

$\int \frac{3 \sin (x)}{- 2 \cos (x)} d x$

Step 5

Separate fractions.

$\int \frac{3}{- 2} \cdot \frac{\sin (x)}{\cos (x)} d x$

Step 6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\int \frac{3}{- 2} \tan (x) d x$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\int - \frac{3}{2} \tan (x) d x$

Step 7.2

Combine $\tan (x)$ and $\frac{3}{2}$ .

$\int - \frac{\tan (x) \cdot 3}{2} d x$

Step 7.3

Move $3$ to the left of $\tan (x)$ .

$\int - \frac{3 \tan (x)}{2} d x$

Step 8

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

$- \int \frac{3 \tan (x)}{2} d x$

Step 9

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

$- (\frac{3}{2} \int \tan (x) d x)$

Step 10

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$- \frac{3}{2} (\ln (| \sec (x) |) + C)$

Step 11

Simplify.

$- \frac{3}{2} \ln (| \sec (x) |) + C$