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

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

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integral of x/(cos^2(1-4x^2))

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

- \frac{1}{8} tan (1 - 4 x^{2}) + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{8 cos ^{2} ( u )} d u

= \int - \frac{1}{8 cos ^{2} ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{8} \cdot \int \frac{1}{cos ^{2} ( u )} d u

Use the common integral:

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

= - \frac{1}{8} tan (u)

Substitute back

u = 1 - 4 x^{2}

= - \frac{1}{8} tan (1 - 4 x^{2})

Add a constant to the solution

= - \frac{1}{8} tan (1 - 4 x^{2}) + C

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