What is the integral of (x^2)/(cos^2(x^3)) ?

The integral of (x^2)/(cos^2(x^3)) is 1/3 tan(x^3)+C

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integral of (x^2)/(cos^2(x^3))

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

\frac{1}{3} tan (x^{3}) + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Apply u-substitution

: \int \frac{u sec ^{2} ( u ^{\frac{3}{2}} )}{2 x} d u

= \int \frac{sec ^{2} ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= \frac{1}{3} tan (u)

Substitute back

u = x^{3}

= \frac{1}{3} tan (x^{3})

Add a constant to the solution

= \frac{1}{3} tan (x^{3}) + C

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What is the integral of (x^2)/(cos^2(x^3)) ?
The integral of (x^2)/(cos^2(x^3)) is 1/3 tan(x^3)+C