What is the integral of (5+x)/(cos^2(x)) ?

The integral of (5+x)/(cos^2(x)) is 5tan(x)+xtan(x)+ln|cos(x)|+C

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integral of (5+x)/(cos^2(x))

\int \frac{5 + x}{cos ^{2} ( x )} d x

5 tan (x) + x tan (x) + ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{5 + x}{cos ^{2} ( x )} d x

Expand

\frac{5 + x}{cos ^{2} ( x )} : \frac{5}{cos ^{2} ( x )} + \frac{x}{cos ^{2} ( x )}

= \int \frac{5}{cos ^{2} ( x )} + \frac{x}{cos ^{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 \frac{5}{cos ^{2} ( x )} d x + \int \frac{x}{cos ^{2} ( x )} d x

\int \frac{5}{cos ^{2} ( x )} d x = 5 tan (x)

\int \frac{x}{cos ^{2} ( x )} d x = x tan (x) + ln ∣ cos (x) ∣

= 5 tan (x) + x tan (x) + ln ∣ cos (x) ∣

Add a constant to the solution

= 5 tan (x) + x tan (x) + ln ∣ cos (x) ∣ + C

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What is the integral of (5+x)/(cos^2(x)) ?
The integral of (5+x)/(cos^2(x)) is 5tan(x)+xtan(x)+ln|cos(x)|+C