What is the integral of (cos^5(5x+1))/(sin^4(5x+1)) ?

The integral of (cos^5(5x+1))/(sin^4(5x+1)) is 1/5 (-1/3 csc^3(5x+1)+2csc(5x+1)+sin(5x+1))+C

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integral of (cos^5(5x+1))/(sin^4(5x+1))

\int \frac{cos ^{5} ( 5 x + 1 )}{sin ^{4} ( 5 x + 1 )} d x

\frac{1}{5} (- \frac{1}{3} csc^{3} (5 x + 1) + 2 csc (5 x + 1) + sin (5 x + 1)) + C

Solution steps

\int \frac{cos ^{5} ( 5 x + 1 )}{sin ^{4} ( 5 x + 1 )} d x

Apply u-substitution

: \int \frac{cos ^{5} ( u )}{5 sin ^{4} ( u )} d u

= \int \frac{cos ^{5} ( u )}{5 sin ^{4} ( u )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{cos ^{5} ( u )}{sin ^{4} ( u )} d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) ^{2} cos ( u )}{sin ^{4} ( u )} d u

Apply u-substitution

: \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{4}} d v

= \frac{1}{5} \cdot \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{4}} d v

Expand

\frac{( 1 - v ^{2} ) ^{2}}{v ^{4}} : \frac{1}{v ^{4}} - \frac{2}{v ^{2}} + 1

= \frac{1}{5} \cdot \int \frac{1}{v ^{4}} - \frac{2}{v ^{2}} + 1 d v

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}{5} (\int \frac{1}{v ^{4}} d v - \int \frac{2}{v ^{2}} d v + \int 1 d v)

\int \frac{1}{v ^{4}} d v = - \frac{1}{3 v ^{3}}

\int \frac{2}{v ^{2}} d v = - \frac{2}{v}

\int 1 d v = v

= \frac{1}{5} (- \frac{1}{3 v ^{3}} - (- \frac{2}{v}) + v)

Substitute back

= \frac{1}{5} (- \frac{1}{3 sin ^{3} ( 5 x + 1 )} - (- \frac{2}{sin ( 5 x + 1 )}) + sin (5 x + 1))

Simplify

\frac{1}{5} (- \frac{1}{3 sin ^{3} ( 5 x + 1 )} - (- \frac{2}{sin ( 5 x + 1 )}) + sin (5 x + 1)) : \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x + 1) + 2 csc (5 x + 1) + sin (5 x + 1))

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x + 1) + 2 csc (5 x + 1) + sin (5 x + 1))

Add a constant to the solution

= \frac{1}{5} (- \frac{1}{3} csc^{3} (5 x + 1) + 2 csc (5 x + 1) + sin (5 x + 1)) + C

y = \frac{( x - 1 )}{x + 1}

- \frac{5}{7 x + 5}

x \to 0 + lim ((1 + \frac{1}{x})^{x})

\int sin (π t) d t

x \to \infty lim (\frac{x}{e})

What is the integral of (cos^5(5x+1))/(sin^4(5x+1)) ?
The integral of (cos^5(5x+1))/(sin^4(5x+1)) is 1/5 (-1/3 csc^3(5x+1)+2csc(5x+1)+sin(5x+1))+C