What is the integral of (tan^4(x))/(sec^5(x)) ?

The integral of (tan^4(x))/(sec^5(x)) is (sin^5(x))/5+C

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integral of (tan^4(x))/(sec^5(x))

\int \frac{tan ^{4} ( x )}{sec ^{5} ( x )} d x

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

Solution steps

\int \frac{tan ^{4} ( x )}{sec ^{5} ( x )} d x

Use the basic trigonometric identity:

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

= \int cos^{5} (x) tan^{4} (x) d x

Rewrite using trig identities

= \int sin^{4} (x) cos (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 = sin (x)

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

Add a constant to the solution

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

y^{^{'}} = 4 y^{2} x

\int 13 tan^{5} (x) d x

\int \frac{t ^{2}}{1 + t ^{3}} d t

\frac{d y}{d x}, x^{4} y - 8 x y + 3 x y^{2} = 9

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