What is the integral of 7tan^4(x)sec^4(x) ?

The integral of 7tan^4(x)sec^4(x) is 7/5 tan^5(x)+tan^7(x)+C

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integral of 7tan^4(x)sec^4(x)

\int 7 tan^{4} (x) sec^{4} (x) d x

\frac{7}{5} tan^{5} (x) + tan^{7} (x) + C

Solution steps

\int 7 tan^{4} (x) sec^{4} (x) d x

Take the constant out:

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

= 7 \cdot \int tan^{4} (x) sec^{4} (x) d x

Rewrite using trig identities

= 7 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) tan^{4} (x) d x

Apply u-substitution

: \int u^{4} (1 + u^{2}) d u

= 7 \cdot \int u^{4} (1 + u^{2}) d u

Expand

u^{4} (1 + u^{2}) : u^{4} + u^{6}

= 7 \cdot \int u^{4} + u^{6} 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

= 7 (\int u^{4} d u + \int u^{6} d u)

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

\int u^{6} d u = \frac{u ^{7}}{7}

= 7 (\frac{u ^{5}}{5} + \frac{u ^{7}}{7})

Substitute back

u = tan (x)

= 7 (\frac{tan ^{5} ( x )}{5} + \frac{tan ^{7} ( x )}{7})

Simplify

7 (\frac{tan ^{5} ( x )}{5} + \frac{tan ^{7} ( x )}{7}) : \frac{7}{5} tan^{5} (x) + tan^{7} (x)

= \frac{7}{5} tan^{5} (x) + tan^{7} (x)

Add a constant to the solution

= \frac{7}{5} tan^{5} (x) + tan^{7} (x) + C

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What is the integral of 7tan^4(x)sec^4(x) ?
The integral of 7tan^4(x)sec^4(x) is 7/5 tan^5(x)+tan^7(x)+C