What is the integral of (7x^6)/(csc(pi+x^7)) ?

The integral of (7x^6)/(csc(pi+x^7)) is x^7sin(pi+x^7)+x^7sin(x^7)+cos(x^7)+C

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integral of (7x^6)/(csc(pi+x^7))

\int \frac{7 x ^{6}}{csc ( π + x ^{7} )} d x

x^{7} sin (π + x^{7}) + x^{7} sin (x^{7}) + cos (x^{7}) + C

Solution steps

\int \frac{7 x ^{6}}{csc ( π + x ^{7} )} d x

Take the constant out:

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

= 7 \cdot \int \frac{x ^{6}}{csc ( π + x ^{7} )} d x

Apply Integration By Parts

: \frac{1}{7} x^{7} sin (π + x^{7}) - \int - x^{13} cos (x^{7}) d x

= 7 (\frac{1}{7} x^{7} sin (π + x^{7}) - \int - x^{13} cos (x^{7}) d x)

\int - x^{13} cos (x^{7}) d x = - \frac{1}{7} (x^{7} sin (x^{7}) + cos (x^{7}))

= 7 (\frac{1}{7} x^{7} sin (π + x^{7}) - (- \frac{1}{7} (x^{7} sin (x^{7}) + cos (x^{7}))))

Simplify

7 (\frac{1}{7} x^{7} sin (π + x^{7}) - (- \frac{1}{7} (x^{7} sin (x^{7}) + cos (x^{7})))) : x^{7} sin (π + x^{7}) + x^{7} sin (x^{7}) + cos (x^{7})

= x^{7} sin (π + x^{7}) + x^{7} sin (x^{7}) + cos (x^{7})

Add a constant to the solution

= x^{7} sin (π + x^{7}) + x^{7} sin (x^{7}) + cos (x^{7}) + C

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What is the integral of (7x^6)/(csc(pi+x^7)) ?
The integral of (7x^6)/(csc(pi+x^7)) is x^7sin(pi+x^7)+x^7sin(x^7)+cos(x^7)+C