What is the integral of (tan^6(x))/(sec^7(x)) ?

The integral of (tan^6(x))/(sec^7(x)) is (sin^7(x))/7+C

integral of (tan^6(x))/(sec^7(x))

\int \frac{tan ^{6} ( x )}{sec ^{7} ( x )} d x

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

Solution steps

\int \frac{tan ^{6} ( x )}{sec ^{7} ( x )} d x

Use the basic trigonometric identity:

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

= \int cos^{7} (x) tan^{6} (x) d x

Rewrite using trig identities

= \int sin^{6} (x) cos (x) d x

Apply u-substitution

= \int u^{6} d u

Apply the Power Rule

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

Substitute back

u = sin (x)

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

Add a constant to the solution

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

\int \frac{( 2 x ^{2} + x ^{2} x - 1 )}{x ^{2}} d x

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

\int (4 x^{3} + 5 x^{2} - 7 x) d x

\int - sin (u) d u

\int \frac{x ^{2}}{( 4 - 9 x ^{2} )} d x

What is the integral of (tan^6(x))/(sec^7(x)) ?
The integral of (tan^6(x))/(sec^7(x)) is (sin^7(x))/7+C