What is the integral of 4/(sec(x)e^{sin(x))} ?

The integral of 4/(sec(x)e^{sin(x))} is -4e^{-sin(x)}+C

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integral of 4/(sec(x)e^{sin(x))}

\int \frac{4}{sec ( x ) e ^{s i n (x)}} d x

- 4 e^{- s i n (x)} + C

Solution steps

\int \frac{4}{sec ( x ) e ^{s i n (x)}} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{sec ( x ) e ^{s i n (x)}} d x

Rewrite using trig identities

= 4 \cdot \int \frac{cos ( x )}{e ^{s i n (x)}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{cos ( x )}{e ^{s i n (x)}} = (cos (x)) e^{- s i n (x)}

= 4 \cdot \int cos (x) e^{- s i n (x)} d x

Apply u-substitution

: \int - e^{u} d u

= 4 \cdot \int - e^{u} d u

Take the constant out:

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

= 4 (- \int e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

= 4 (- e^{u})

Substitute back

u = - sin (x)

= 4 (- e^{- s i n (x)})

Simplify

= - 4 e^{- s i n (x)}

Add a constant to the solution

= - 4 e^{- s i n (x)} + C

y = (1 + ln (x))^{s i n (x)}

\frac{d}{d x} ((5 x)^{\frac{1}{3}})

\int_{0}^{8} x (\frac{1}{8} x) d x

\frac{d}{d x} ((x^{3} - 5)^{5} (x^{5} + 7))

\frac{2 x - 1}{2 ( x ^{2} - x )}

What is the integral of 4/(sec(x)e^{sin(x))} ?
The integral of 4/(sec(x)e^{sin(x))} is -4e^{-sin(x)}+C