What is the integral of xsec^2(3-4x^2) ?

The integral of xsec^2(3-4x^2) is -1/8 tan(3-4x^2)+C

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integral of xsec^2(3-4x^2)

\int x sec^{2} (3 - 4 x^{2}) d x

- \frac{1}{8} tan (3 - 4 x^{2}) + C

Solution steps

\int x sec^{2} (3 - 4 x^{2}) d x

Rewrite using trig identities

= \int \frac{x}{cos ^{2} ( 3 - 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 = 3 - 4 x^{2}

= - \frac{1}{8} tan (3 - 4 x^{2})

Add a constant to the solution

= - \frac{1}{8} tan (3 - 4 x^{2}) + C

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What is the integral of xsec^2(3-4x^2) ?
The integral of xsec^2(3-4x^2) is -1/8 tan(3-4x^2)+C