What is the integral of (5x+3-xsec^2(x))/x ?

The integral of (5x+3-xsec^2(x))/x is 5x+3ln|x|-tan(x)+C

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integral of (5x+3-xsec^2(x))/x

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

5 x + 3 ln ∣ x ∣ - tan (x) + C

Solution steps

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

Expand

\frac{5 x + 3 - x sec ^{2} ( x )}{x} : 5 + \frac{3}{x} - sec^{2} (x)

= \int 5 + \frac{3}{x} - sec^{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 5 d x + \int \frac{3}{x} d x - \int sec^{2} (x) d x

\int 5 d x = 5 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= 5 x + 3 ln ∣ x ∣ - tan (x)

Add a constant to the solution

= 5 x + 3 ln ∣ x ∣ - tan (x) + C

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What is the integral of (5x+3-xsec^2(x))/x ?
The integral of (5x+3-xsec^2(x))/x is 5x+3ln|x|-tan(x)+C