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

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

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integral of (tan^3(x)+1)/(cos^2(x))

\int \frac{tan ^{3} ( x ) + 1}{cos ^{2} ( x )} d x

\frac{tan ^{4} ( x )}{4} + tan (x) + C

Solution steps

\int \frac{tan ^{3} ( x ) + 1}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

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

Apply u-substitution

= \int u^{3} + 1 d u

Apply the Sum Rule:

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

= \int u^{3} d u + \int 1 d u

\int u^{3} d u = \frac{u ^{4}}{4}

\int 1 d u = u

= \frac{u ^{4}}{4} + u

Substitute back

u = tan (x)

= \frac{tan ^{4} ( x )}{4} + tan (x)

Add a constant to the solution

= \frac{tan ^{4} ( x )}{4} + tan (x) + C

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