What is the integral of (tan(x))^2sec(x) ?

The integral of (tan(x))^2sec(x) is -xsec^3(x)cos^3(x)+x+1/2 sec(x)tan(x)-1/2 ln|tan(x)+sec(x)|+C

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integral of (tan(x))^2sec(x)

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

- x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{1 - cos ^{2} ( x )}{cos ^{3} ( x )} : \frac{1}{cos ^{3} ( x )} - \frac{1}{cos ( x )}

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

\int \frac{1}{cos ^{3} ( x )} d x = - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

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

Add similar elements:

\frac{1}{2} ln ∣ tan (x) + sec (x) ∣ - ln ∣ tan (x) + sec (x) ∣ = - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

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

Add a constant to the solution

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

\frac{\partial}{\partial x} (6 - x^{3} - y^{5})

\frac{d}{d x} (arcsec (7 x))

\frac{2 x ^{2} + 4 x}{( x + 1 ) ^{2}}

\frac{d y}{d x} + \frac{1}{x} y = \frac{1}{y}

\int_{- \infty}^{- 4} x e^{x} d x

What is the integral of (tan(x))^2sec(x) ?
The integral of (tan(x))^2sec(x) is -xsec^3(x)cos^3(x)+x+1/2 sec(x)tan(x)-1/2 ln|tan(x)+sec(x)|+C