Calculus Examples

$\int (5 \sec (x) \tan (x) - \frac{3}{2} \csc^{2} (x) + 9) d x$

Step 1

Remove parentheses.

$\int 5 \sec (x) \tan (x) - \frac{3}{2} \cdot \csc^{2} (x) + 9 d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 \sec (x) \tan (x) d x + \int - \frac{3}{2} \cdot \csc^{2} (x) d x + \int 9 d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sec (x) \tan (x) d x + \int - \frac{3}{2} \cdot \csc^{2} (x) d x + \int 9 d x$

Step 4

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$5 (\sec (x) + C) + \int - \frac{3}{2} \cdot \csc^{2} (x) d x + \int 9 d x$

Step 5

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

$5 (\sec (x) + C) - \frac{3}{2} \int \csc^{2} (x) d x + \int 9 d x$

Step 6

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$5 (\sec (x) + C) - \frac{3}{2} (- \cot (x) + C) + \int 9 d x$

Step 7

Apply the constant rule.

$5 (\sec (x) + C) - \frac{3}{2} (- \cot (x) + C) + 9 x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 \sec (x) + \frac{3 \cot (x)}{2} + 9 x + C$

Step 8.2

Reorder terms.

$5 \sec (x) + \frac{3}{2} \cot (x) + 9 x + C$