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

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

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integral of (sin(x))/(cos^7(x))

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

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

Solution steps

\int \frac{sin ( x )}{cos ^{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

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \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

y^{^{'}} + 8 y - 6 = 0

\int r 1 + r^{2} d r

3 x^{2} e^{x} + e^{x} (x^{3} + 1)

\int - 4 x d x

\frac{d y}{d x} = \frac{y ( - 2 x + 4 )}{( x ^{2} + 1 ) ( x - 1 ) ^{2}}

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