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

The integral of (sin(x)+tan(x))/(cos^3(x)) is (tan^2(x))/2+1/3 sec^3(x)+C

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

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

\frac{tan ^{2} ( x )}{2} + \frac{1}{3} sec^{3} (x) + C

Solution steps

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

Expand

\frac{sin ( x ) + tan ( x )}{cos ^{3} ( x )} : \frac{sin ( x )}{cos ^{3} ( x )} + \frac{tan ( x )}{cos ^{3} ( x )}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{sin ( x )}{cos ^{3} ( x )} d x = \frac{tan ^{2} ( x )}{2}

\int \frac{tan ( x )}{cos ^{3} ( x )} d x = \frac{1}{3} sec^{3} (x)

= \frac{tan ^{2} ( x )}{2} + \frac{1}{3} sec^{3} (x)

Add a constant to the solution

= \frac{tan ^{2} ( x )}{2} + \frac{1}{3} sec^{3} (x) + C

What is the integral of (sin(x)+tan(x))/(cos^3(x)) ?
The integral of (sin(x)+tan(x))/(cos^3(x)) is (tan^2(x))/2+1/3 sec^3(x)+C