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

The integral of (tan^4(x))/(cos^2(x)) is (tan^5(x))/5+C

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integral of (tan^4(x))/(cos^2(x))

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

\frac{tan ^{5} ( x )}{5} + C

Solution steps

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

Use the basic trigonometric identity:

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

= \int tan^{4} (x) sec^{2} (x) d x

Rewrite using trig identities

= \int (1 + tan^{2} (x)) tan^{4} (x) d x

Apply u-substitution

: \int u^{4} d u

= \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{u ^{5}}{5}

Substitute back

u = tan (x)

= \frac{tan ^{5} ( x )}{5}

Add a constant to the solution

= \frac{tan ^{5} ( x )}{5} + C

\frac{d y}{d x} = y e^{x}, y (0) = 2 e

\frac{d}{d x} (sin (y x))

\frac{d}{d t} (1 + t^{2})

\int_{0}^{1} (1 - x) d x

What is the integral of (tan^4(x))/(cos^2(x)) ?
The integral of (tan^4(x))/(cos^2(x)) is (tan^5(x))/5+C