What is the integral of 7tan(x) ?

The integral of 7tan(x) is -7ln|cos(x)|+C

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integral of 7tan(x)

\int 7 tan (x) d x

- 7 ln ∣ cos (x) ∣ + C

Solution steps

\int 7 tan (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 (x) d x

Rewrite using trig identities

= 7 \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int - \frac{1}{u} d u

= 7 \cdot \int - \frac{1}{u} d u

Take the constant out:

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

= 7 (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 (- ln ∣ u ∣)

Substitute back

u = cos (x)

= 7 (- ln ∣ cos (x) ∣)

Simplify

= - 7 ln ∣ cos (x) ∣

Add a constant to the solution

= - 7 ln ∣ cos (x) ∣ + C

\int \frac{1}{sin ^{5} ( x ) cos ( x )} d x

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

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (0) = 1, y^{^{'}} (0) = 0

\int_{\frac{1}{2}}^{1} (x^{- 3} - 25) d x

y^{^{''}} + \frac{1}{x} y^{^{'}} = 0