What is the integral of tan^2(x)(tan^2(x)+1)sec^2(x) ?

The integral of tan^2(x)(tan^2(x)+1)sec^2(x) is (tan^5(x))/5+(tan^3(x))/3+C

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integral of tan^2(x)(tan^2(x)+1)sec^2(x)

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

\frac{tan ^{5} ( x )}{5} + \frac{tan ^{3} ( x )}{3} + C

Solution steps

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

Apply u-substitution

: \int u^{2} (u^{2} + 1) d u

= \int u^{2} (u^{2} + 1) d u

Expand

u^{2} (u^{2} + 1) : u^{4} + u^{2}

= \int u^{4} + u^{2} 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^{4} d u + \int u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{2} d u = \frac{u ^{3}}{3}

= \frac{u ^{5}}{5} + \frac{u ^{3}}{3}

Substitute back

u = tan (x)

= \frac{tan ^{5} ( x )}{5} + \frac{tan ^{3} ( x )}{3}

Add a constant to the solution

= \frac{tan ^{5} ( x )}{5} + \frac{tan ^{3} ( x )}{3} + C

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What is the integral of tan^2(x)(tan^2(x)+1)sec^2(x) ?
The integral of tan^2(x)(tan^2(x)+1)sec^2(x) is (tan^5(x))/5+(tan^3(x))/3+C