What is the integral of (x^3)/(x^4cos^2(ln(x))) ?

The integral of (x^3)/(x^4cos^2(ln(x))) is tan(ln(x))+C

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integral of (x^3)/(x^4cos^2(ln(x)))

\int \frac{x ^{3}}{x ^{4} cos ^{2} ( ln ( x ) )} d x

tan (ln (x)) + C

Solution steps

\int \frac{x ^{3}}{x ^{4} cos ^{2} ( ln ( x ) )} d x

Simplify

\frac{x ^{3}}{x ^{4} cos ^{2} ( ln ( x ) )} : \frac{1}{x cos ^{2} ( ln ( x ) )}

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

Apply u-substitution

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

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

Use the common integral:

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

= tan (u)

Substitute back

u = ln (x)

= tan (ln (x))

Add a constant to the solution

= tan (ln (x)) + C

\int (\frac{x ^{3} + 5 x ^{2} - 4}{x ^{2}}) d x

\int \frac{( ln ^{5} ( x ) )}{x} d x

\int \frac{e ^{- l n (x)}}{x} d x

\int - 3 lo g_{3} (x) d x

\int e^{- \frac{2 x}{a}} d x

What is the integral of (x^3)/(x^4cos^2(ln(x))) ?
The integral of (x^3)/(x^4cos^2(ln(x))) is tan(ln(x))+C