Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \cos (3 z+4) d z$$

Short answer

The integral of \(\cos (3z+4)\) dz is \(1/3 * \sin(3z+4) + C\).

Step-by-step solution

Identify the substitution

In this exercise, the substitution that will turn the integral into a simpler form is \(u = 3z+4\). With this substitution, the integral \(\int \cos (3 z+4) d z\) becomes \(\int \cos (u) d u\), once the derivative of \(u\) with respect to \(z\) is also taken into account.

Find the differential du

In order to replace the \(dz\) in the integral, it is necessary to find \(du\) in terms of \(dz\). The derivative of \(u = 3z+4\) with respect to \(z\) is \(du/dz = 3\). Therefore, \(du = 3dz\), and solve it for \(dz\), it will become \(dz = du/3\).

Substitute into the integral

Now that the substitution and differential are established, they can both be substituted into the original integral: \(\int \cos (3 z+4) d z\) becomes \(\int \cos(u) * (du/3)\). The 1/3 can be taken outside the integral, leaving \(1/3 \int \cos(u) du\).

Evaluate the integral

The integral of \(\cos(u)\) is \(\sin(u)\), hence, \(1/3 \int \cos(u) du = 1/3 * \sin(u) + C\), where C is the constant of integration.

Substitute u back into the solution

Finally, to complete the solution, substitute the original \(u = 3z+4\) back into the equation, yielding \(1/3 * \sin(3z+4) + C\).