What is the integral of tsec^2(t^2)tan^4(t^2) ?

The integral of tsec^2(t^2)tan^4(t^2) is 1/10 tan^5(t^2)+C

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integral of tsec^2(t^2)tan^4(t^2)

\int t sec^{2} (t^{2}) tan^{4} (t^{2}) d t

\frac{1}{10} tan^{5} (t^{2}) + C

Solution steps

\int t sec^{2} (t^{2}) tan^{4} (t^{2}) d t

Apply u-substitution

: \int \frac{tan ^{4} ( u ) sec ^{2} ( u )}{2} d u

= \int \frac{tan ^{4} ( u ) sec ^{2} ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int tan^{4} (u) sec^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int (1 + tan^{2} (u)) tan^{4} (u) d u

Apply u-substitution

: \int v^{4} d v

= \frac{1}{2} \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= \frac{1}{2} \cdot \frac{v ^{5}}{5}

Substitute back

= \frac{1}{2} \cdot \frac{tan ^{5} ( t ^{2} )}{5}

Simplify

\frac{1}{2} \cdot \frac{tan ^{5} ( t ^{2} )}{5} : \frac{1}{10} tan^{5} (t^{2})

= \frac{1}{10} tan^{5} (t^{2})

Add a constant to the solution

= \frac{1}{10} tan^{5} (t^{2}) + C

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What is the integral of tsec^2(t^2)tan^4(t^2) ?
The integral of tsec^2(t^2)tan^4(t^2) is 1/10 tan^5(t^2)+C