Question

match each indefinite integral to its result, where ( c ) is a constant.
integral | result
( int 56cos(8z - 5),dz ) | ( 8cos(7z - 5) + c )
( int -56sin(7z - 5),dz ) | ( 7sin(8z - 5) + c )
( int 56cos(7z - 5),dz ) | ( 8sin(7z - 5) + c )

Explanation:

Step1: Integrate first integral

Use substitution: let $u=8x-5$, $du=8dx$, so $dx=\frac{du}{8}$.
$$\int 56\cos(8x-5)dx = 56\cdot\frac{1}{8}\int\cos(u)du = 7\sin(u)+C = 7\sin(8x-5)+C$$

Step2: Integrate second integral

Use substitution: let $u=7x-5$, $du=7dx$, so $dx=\frac{du}{7}$.
$$\int -56\sin(7x-5)dx = -56\cdot\frac{1}{7}\int\sin(u)du = 8\cos(u)+C = 8\cos(7x-5)+C$$

Step3: Integrate third integral

Use substitution: let $u=7x-5$, $du=7dx$, so $dx=\frac{du}{7}$.
$$\int 56\cos(7x-5)dx = 56\cdot\frac{1}{7}\int\cos(u)du = 8\sin(u)+C = 8\sin(7x-5)+C$$

Answer:

1. $\boldsymbol{\int 56\cos(8x - 5)dx}$ matches $\boldsymbol{7\sin(8x - 5) + C}$
2. $\boldsymbol{\int -56\sin(7x - 5)dx}$ matches $\boldsymbol{8\cos(7x - 5) + C}$
3. $\boldsymbol{\int 56\cos(7x - 5)dx}$ matches $\boldsymbol{8\sin(7x - 5) + C}$