What is the integral of x^e+2/x+sec^2(2x) ?

The integral of x^e+2/x+sec^2(2x) is (x^{e+1})/(e+1)+2ln|x|+1/2 tan(2x)+C

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integral of x^e+2/x+sec^2(2x)

\int x^{e} + \frac{2}{x} + sec^{2} (2 x) d x

\frac{x ^{e + 1}}{e + 1} + 2 ln ∣ x ∣ + \frac{1}{2} tan (2 x) + C

Solution steps

\int x^{e} + \frac{2}{x} + sec^{2} (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 x^{e} d x + \int \frac{2}{x} d x + \int sec^{2} (2 x) d x

\int x^{e} d x = \frac{x ^{e + 1}}{e + 1}

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int sec^{2} (2 x) d x = \frac{1}{2} tan (2 x)

= \frac{x ^{e + 1}}{e + 1} + 2 ln ∣ x ∣ + \frac{1}{2} tan (2 x)

Add a constant to the solution

= \frac{x ^{e + 1}}{e + 1} + 2 ln ∣ x ∣ + \frac{1}{2} tan (2 x) + C

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What is the integral of x^e+2/x+sec^2(2x) ?
The integral of x^e+2/x+sec^2(2x) is (x^{e+1})/(e+1)+2ln|x|+1/2 tan(2x)+C