What is the integral of (sin(x))/(cos^3(x)) ?

The integral of (sin(x))/(cos^3(x)) is (tan^2(x))/2+C

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integral of (sin(x))/(cos^3(x))

integral

\frac{sin ( x )}{cos ^{3} ( x )}

\frac{tan ^{2} ( x )}{2} + C

Solution steps

\int \frac{sin ( x )}{cos ^{3} ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{u ^{2}}{2}

Substitute back

u = tan (x)

= \frac{tan ^{2} ( x )}{2}

Add a constant to the solution

= \frac{tan ^{2} ( x )}{2} + C

\frac{2 x}{( 1 - x ^{2} )}

ln (3 x - 2)

\frac{( 1 + 2 x )}{( 1 + x )}

\frac{1}{( x ^{2} - 9 ) ^{2}}

\frac{9}{x ^{5}}

What is the integral of (sin(x))/(cos^3(x)) ?
The integral of (sin(x))/(cos^3(x)) is (tan^2(x))/2+C