What is the integral of 5^x(2-cot(5^x)(1-csc(5^x))) ?

The integral of 5^x(2-cot(5^x)(1-csc(5^x))) is 1/(ln(5))(2*5^x+ln|csc(5^x)|-csc(5^x))+C

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integral of 5^x(2-cot(5^x)(1-csc(5^x)))

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

\frac{1}{ln ( 5 )} (2 \cdot 5^{x} + ln ∣ csc (5^{x}) ∣ - csc (5^{x})) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 - cot ( u ) ( 1 - csc ( u ) )}{ln ( 5 )} d u

= \int \frac{2 - cot ( u ) ( 1 - csc ( u ) )}{ln ( 5 )} d u

Take the constant out:

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

= \frac{1}{ln ( 5 )} \cdot \int 2 - cot (u) (1 - csc (u)) 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

= \frac{1}{ln ( 5 )} (\int 2 d u - \int cot (u) (1 - csc (u)) d u)

\int 2 d u = 2 u

\int cot (u) (1 - csc (u)) d u = - ln ∣ csc (u) ∣ + csc (u)

= \frac{1}{ln ( 5 )} (2 u - (- ln ∣ csc (u) ∣ + csc (u)))

Substitute back

u = 5^{x}

= \frac{1}{ln ( 5 )} (2 \cdot 5^{x} - (- ln ∣ csc (5^{x}) ∣ + csc (5^{x})))

Simplify

= \frac{1}{ln ( 5 )} (2 \cdot 5^{x} + ln ∣ csc (5^{x}) ∣ - csc (5^{x}))

Add a constant to the solution

= \frac{1}{ln ( 5 )} (2 \cdot 5^{x} + ln ∣ csc (5^{x}) ∣ - csc (5^{x})) + C

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What is the integral of 5^x(2-cot(5^x)(1-csc(5^x))) ?
The integral of 5^x(2-cot(5^x)(1-csc(5^x))) is 1/(ln(5))(2*5^x+ln|csc(5^x)|-csc(5^x))+C