What is the integral of 7xcsc(x^2) ?

The integral of 7xcsc(x^2) is 7/2 ln|tan((x^2)/2)|+C

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integral of 7xcsc(x^2)

\int 7 x csc (x^{2}) d x

\frac{7}{2} ln tan (\frac{x ^{2}}{2}) + C

Solution steps

\int 7 x csc (x^{2}) d x

Take the constant out:

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

= 7 \cdot \int x csc (x^{2}) d x

Apply u-substitution

: \int \frac{csc ( u )}{2} d u

= 7 \cdot \int \frac{csc ( u )}{2} d u

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 csc (u) d u

Apply u-substitution

: \int \frac{1}{v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 7 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 7 \cdot \frac{1}{2} ln tan (\frac{x ^{2}}{2})

Simplify

7 \cdot \frac{1}{2} ln tan (\frac{x ^{2}}{2}) : \frac{7}{2} ln tan (\frac{x ^{2}}{2})

= \frac{7}{2} ln tan (\frac{x ^{2}}{2})

Add a constant to the solution

= \frac{7}{2} ln tan (\frac{x ^{2}}{2}) + C

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What is the integral of 7xcsc(x^2) ?
The integral of 7xcsc(x^2) is 7/2 ln|tan((x^2)/2)|+C