What is the integral of sin(x)sec^7(x) ?

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

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integral of sin(x)sec^7(x)

\int sin (x) sec^{7} (x) d x

\frac{sec ^{6} ( x )}{6} + C

Solution steps

\int sin (x) sec^{7} (x) d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Apply u-substitution

= \int u^{5} d u

Apply the Power Rule

= \frac{u ^{6}}{6}

Substitute back

u = sec (x)

= \frac{sec ^{6} ( x )}{6}

Add a constant to the solution

= \frac{sec ^{6} ( x )}{6} + C

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