What is the integral of (sin^2(x)+1)/(cos^4(x)) ?

The integral of (sin^2(x)+1)/(cos^4(x)) is 2/3 tan^3(x)+tan(x)+C

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integral of (sin^2(x)+1)/(cos^4(x))

\int \frac{sin ^{2} ( x ) + 1}{cos ^{4} ( x )} d x

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

Solution steps

\int \frac{sin ^{2} ( x ) + 1}{cos ^{4} ( x )} d x

Expand

\frac{sin ^{2} ( x ) + 1}{cos ^{4} ( x )} : \frac{sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{1}{cos ^{4} ( x )}

= \int \frac{sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{1}{cos ^{4} ( 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 ^{2} ( x )}{cos ^{4} ( x )} d x + \int \frac{1}{cos ^{4} ( x )} d x

\int \frac{sin ^{2} ( x )}{cos ^{4} ( x )} d x = \frac{tan ^{3} ( x )}{3}

\int \frac{1}{cos ^{4} ( x )} d x = tan (x) + \frac{tan ^{3} ( x )}{3}

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

Simplify

\frac{tan ^{3} ( x )}{3} + tan (x) + \frac{tan ^{3} ( x )}{3} : \frac{2}{3} tan^{3} (x) + tan (x)

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

Add a constant to the solution

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

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What is the integral of (sin^2(x)+1)/(cos^4(x)) ?
The integral of (sin^2(x)+1)/(cos^4(x)) is 2/3 tan^3(x)+tan(x)+C