Find the indefinite integral. $\int \csc ( 4 x ) d x$
A) $\frac { - 1 } { 4 } \ln | \csc x + \cot x | + C$
B) $\frac { - 1 } { 4 } \ln | \csc ( 4 x ) + \cot ( 4 x ) | + C$
C) $4 \ln | \csc ( 4 x ) + \cot ( 4 x ) | + C$
D) $\frac { - 1 } { 4 } \ln | \cot ( 4 x ) | + C$
E) $\frac { - 1 } { 4 } \ln | \csc ( 4 x ) | + C$

Question

Quizplus · Accepted Answer

We can start with the substitution $u = 4x$ and $du = 4dx$. Then the integral becomes
$$\int \csc(4x)dx = \frac{1}{4}\int \frac{1}{\sin(u)}du = \frac{1}{4}\ln|\csc(u) + \cot(u)| + C.$$
Substituting back for $u=4x$ gives us
$$\int \csc(4x)dx = \frac{1}{4}\ln|\csc(4x) + \cot(4x)| + C,$$
which matches option B.

Find the Indefinite Integral ∫csc⁡(4x)dx\int \csc ( 4 x ) d x∫csc(4x)dx

Find the Indefinite Integral $\int \csc ( 4 x ) d x$