What is the integral of (tan^4(x))/(sec^7(x)) ?

The integral of (tan^4(x))/(sec^7(x)) is (sin^5(x))/5-(sin^7(x))/7+C

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integral of (tan^4(x))/(sec^7(x))

\int \frac{tan ^{4} ( x )}{sec ^{7} ( x )} d x

\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

Solution steps

\int \frac{tan ^{4} ( x )}{sec ^{7} ( x )} d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) sin^{4} (x) d x

Apply u-substitution

: \int u^{4} (1 - u^{2}) d u

= \int u^{4} (1 - u^{2}) d u

Expand

u^{4} (1 - u^{2}) : u^{4} - u^{6}

= \int u^{4} - u^{6} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{4} d u - \int u^{6} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{u ^{5}}{5} - \frac{u ^{7}}{7}

Substitute back

u = sin (x)

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}

Add a constant to the solution

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

\int 3 sin^{2} (t) cos (t) d t

\int x^{4} e^{3 x} d x

\frac{d}{d x} (2 sec (2 x) tan (2 x))

0.125 x 64 - x^{2}

\frac{\partial}{\partial x} (x^{2} + sin (x y))

What is the integral of (tan^4(x))/(sec^7(x)) ?
The integral of (tan^4(x))/(sec^7(x)) is (sin^5(x))/5-(sin^7(x))/7+C