What is the integral of sec^2(x)(1-4x) ?

The integral of sec^2(x)(1-4x) is tan(x)-8(1/2 xtan(x)+1/2 ln|cos(x)|)+C

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integral of sec^2(x)(1-4x)

\int sec^{2} (x) (1 - 4 x) d x

tan (x) - 8 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) + C

Solution steps

\int sec^{2} (x) (1 - 4 x) d x

Rewrite using trig identities

= \int \frac{1 - 4 x}{cos ^{2} ( x )} d x

Expand

\frac{1 - 4 x}{cos ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} - \frac{4 x}{cos ^{2} ( x )}

= \int \frac{1}{cos ^{2} ( x )} - \frac{4 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{1}{cos ^{2} ( x )} d x - \int \frac{4 x}{cos ^{2} ( x )} d x

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

\int \frac{4 x}{cos ^{2} ( x )} d x = 8 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣)

= tan (x) - 8 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣)

Add a constant to the solution

= tan (x) - 8 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) + C

What is the integral of sec^2(x)(1-4x) ?
The integral of sec^2(x)(1-4x) is tan(x)-8(1/2 xtan(x)+1/2 ln|cos(x)|)+C