Problem 852

Evaluate the indefinite integral by using the given substitution to reduce the integral to standard from.

∫ 6sec²(3x)tan (3x)dx,

a. u = tan(3x)

b. u = sec(3x)

Solution:-

The Substitution Rule state if u = f(x) is a differentiable function whose range is an interval I and f is continuous on I, then

∫ f(g(x))g’(x)dx = ∫f(u)du.

Substitute u for g(x) and du/g’(x) for dx to obtain the integral ∫ f(u)du.

For part a, the function u = g(x) = tan(3x) is given.

Thus , du = 3sec²(3x)dx, or sec²(3x)dx = $\frac{1}{3}$ du.

Integrate with respect to u.

∫6u. $\frac{1}{3}$ du = u²+C

Replacing u by tan (3x), ∫6sec²(3x)tan(3x)dx = tan²(3x)+C.

For part b, the function u = g(x)= sec(3x)is given.

Thus , du = 3sec(3x)tan(3x)dx, or sec(3x)tan(3x)dx = $\frac{1}{3}$ du.

Integrate with respect to u.

∫6u. $\frac{1}{3}$ du= u²+C

Replacing u by sec(3x), ∫6 sec²(3x)tan(3x)dx = sec²(3x)+C.