What is the integral of 7/(tan(t)(csc(t)+4sin(t))) ?

The integral of 7/(tan(t)(csc(t)+4sin(t))) is 7/2 arctan(2sin(t))+C

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integral of 7/(tan(t)(csc(t)+4sin(t)))

\int \frac{7}{tan ( t ) ( csc ( t ) + 4 sin ( t ) )} d t

\frac{7}{2} arctan (2 sin (t)) + C

Solution steps

\int \frac{7}{tan ( t ) ( csc ( t ) + 4 sin ( t ) )} d t

Take the constant out:

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

= 7 \cdot \int \frac{1}{tan ( t ) ( csc ( t ) + 4 sin ( t ) )} d t

Rewrite using trig identities

= 7 \cdot \int \frac{cos ( t )}{1 + 4 sin ^{2} ( t )} d t

Apply u-substitution

= 7 \cdot \int \frac{1}{1 + 4 u ^{2}} d u

Apply Integral Substitution

= 7 \cdot \int \frac{1}{2 ( 1 + v ^{2} )} d v

Take the constant out:

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

= 7 \cdot \frac{1}{2} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= 7 \cdot \frac{1}{2} arctan (v)

Substitute back

= 7 \cdot \frac{1}{2} arctan (2 sin (t))

Simplify

7 \cdot \frac{1}{2} arctan (2 sin (t)) : \frac{7}{2} arctan (2 sin (t))

= \frac{7}{2} arctan (2 sin (t))

Add a constant to the solution

= \frac{7}{2} arctan (2 sin (t)) + C

What is the integral of 7/(tan(t)(csc(t)+4sin(t))) ?
The integral of 7/(tan(t)(csc(t)+4sin(t))) is 7/2 arctan(2sin(t))+C