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

The integral of tan(x)+3/2-3sin^2(x) is -ln|cos(x)|+3/4 sin(2x)+C

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integral of tan(x)+3/2-3sin^2(x)

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

- ln ∣ cos (x) ∣ + \frac{3}{4} sin (2 x) + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int tan (x) d x + \int \frac{3}{2} d x - \int 3 sin^{2} (x) d x

\int tan (x) d x = - ln ∣ cos (x) ∣

\int \frac{3}{2} d x = \frac{3}{2} x

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

= - ln ∣ cos (x) ∣ + \frac{3}{2} x - \frac{3}{2} (x - \frac{1}{2} sin (2 x))

Simplify

- ln ∣ cos (x) ∣ + \frac{3}{2} x - \frac{3}{2} (x - \frac{1}{2} sin (2 x)) : - ln ∣ cos (x) ∣ + \frac{3}{4} sin (2 x)

= - ln ∣ cos (x) ∣ + \frac{3}{4} sin (2 x)

Add a constant to the solution

= - ln ∣ cos (x) ∣ + \frac{3}{4} sin (2 x) + C

\frac{d}{d x} (x - cos (x))

\frac{d}{d x} ((x + 1) (x^{3} + x - 2))

\int \frac{5}{( x - 1 ) ( x ^{2} + 4 )} d x

2 x + 1, (1, 3)

8 x^{7} y + x^{8} y^{^{'}} = - 5 sec^{2} (x) tan (x)

What is the integral of tan(x)+3/2-3sin^2(x) ?
The integral of tan(x)+3/2-3sin^2(x) is -ln|cos(x)|+3/4 sin(2x)+C