What is the integral of x^2tan(x^3)ln(sec^2(x^3)) ?

The integral of x^2tan(x^3)ln(sec^2(x^3)) is 1/12 ln^2(sec^2(x^3))+C

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integral of x^2tan(x^3)ln(sec^2(x^3))

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

\frac{1}{12} ln^{2} (sec^{2} (x^{3})) + C

Solution steps

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

Apply u-substitution

: \int \frac{ln ( u )}{6 u} d u

= \int \frac{ln ( u )}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{6} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{6} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{6} \cdot \frac{ln ^{2} ( sec ^{2} ( x ^{3} ) )}{2}

Simplify

\frac{1}{6} \cdot \frac{ln ^{2} ( sec ^{2} ( x ^{3} ) )}{2} : \frac{1}{12} ln^{2} (sec^{2} (x^{3}))

= \frac{1}{12} ln^{2} (sec^{2} (x^{3}))

Add a constant to the solution

= \frac{1}{12} ln^{2} (sec^{2} (x^{3})) + C

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What is the integral of x^2tan(x^3)ln(sec^2(x^3)) ?
The integral of x^2tan(x^3)ln(sec^2(x^3)) is 1/12 ln^2(sec^2(x^3))+C