What is the integral of-sec^2((3x)/2) ?

The integral of-sec^2((3x)/2) is -2/3 tan((3x)/2)+C

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integral of-sec^2((3x)/2)

\int - sec^{2} (\frac{3 x}{2}) d x

- \frac{2}{3} tan (\frac{3 x}{2}) + C

Solution steps

\int - sec^{2} (\frac{3 x}{2}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int sec^{2} (\frac{3 x}{2}) d x

Apply u-substitution

= - \int sec^{2} (u) \frac{2}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{2}{3} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= - \frac{2}{3} tan (u)

Substitute back

u = \frac{3 x}{2}

= - \frac{2}{3} tan (\frac{3 x}{2})

Add a constant to the solution

= - \frac{2}{3} tan (\frac{3 x}{2}) + C

What is the integral of-sec^2((3x)/2) ?
The integral of-sec^2((3x)/2) is -2/3 tan((3x)/2)+C