What is the integral of 51sec^4(x) ?

The integral of 51sec^4(x) is 51(tan(x)+(tan^3(x))/3)+C

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integral of 51sec^4(x)

\int 51 sec^{4} (x) d x

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

Solution steps

\int 51 sec^{4} (x) d x

Take the constant out:

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

= 51 \cdot \int sec^{4} (x) d x

Rewrite using trig identities

= 51 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) d x

Apply u-substitution

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

= 51 \cdot \int 1 + 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

= 51 (\int 1 d u + \int u^{2} d u)

\int 1 d u = u

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

= 51 (u + \frac{u ^{3}}{3})

Substitute back

u = tan (x)

= 51 (tan (x) + \frac{tan ^{3} ( x )}{3})

Add a constant to the solution

= 51 (tan (x) + \frac{tan ^{3} ( x )}{3}) + C

\int \frac{2 x}{( x - 1 ) ^{5}} d x

\frac{d}{d x} (\frac{3}{1 - x})

f (t) = ln (t)

\int 2^{s i n (x)} cos (x) d x

0.145 \cdot \frac{d v}{d t} = (0.145 \cdot 9.81) - (0.043 \cdot v)

What is the integral of 51sec^4(x) ?
The integral of 51sec^4(x) is 51(tan(x)+(tan^3(x))/3)+C