What is the integral of sin(x)sec^8(x) ?

The integral of sin(x)sec^8(x) is (sec^7(x))/7+C

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integral of sin(x)sec^8(x)

\int sin (x) sec^{8} (x) d x

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

Solution steps

\int sin (x) sec^{8} (x) d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Apply u-substitution

: \int u^{6} d u

= \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

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

Substitute back

u = sec (x)

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

Add a constant to the solution

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

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

\frac{d}{d x} (9 tan (x) sec (x))

\int \frac{1}{1 + 3 x} d x

\frac{5}{4 + x ^{2}}, at x = 2

\int 3 csc^{2} (x) d x

What is the integral of sin(x)sec^8(x) ?
The integral of sin(x)sec^8(x) is (sec^7(x))/7+C