What is the integral of 10sec^2(x)tan^4(x) ?

The integral of 10sec^2(x)tan^4(x) is 2tan^5(x)+C

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integral of 10sec^2(x)tan^4(x)

\int 10 sec^{2} (x) tan^{4} (x) d x

2 tan^{5} (x) + C

Solution steps

\int 10 sec^{2} (x) tan^{4} (x) d x

Take the constant out:

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

= 10 \cdot \int sec^{2} (x) tan^{4} (x) d x

Rewrite using trig identities

= 10 \cdot \int (1 + tan^{2} (x)) tan^{4} (x) d x

Apply u-substitution

: \int u^{4} d u

= 10 \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= 10 \cdot \frac{u ^{5}}{5}

Substitute back

u = tan (x)

= 10 \cdot \frac{tan ^{5} ( x )}{5}

Simplify

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

= 2 tan^{5} (x)

Add a constant to the solution

= 2 tan^{5} (x) + C

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What is the integral of 10sec^2(x)tan^4(x) ?
The integral of 10sec^2(x)tan^4(x) is 2tan^5(x)+C