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

∫ 6sec^{2}(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 = 3sec^{2}(3x)dx, or sec^{2}(3x)dx = du.

Integrate with respect to u.

∫6u. du = u^{2}+C

Replacing u by tan (3x), ∫6sec^{2}(3x)tan(3x)dx = tan^{2}(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 = du.

Integrate with respect to u.

∫6u. du= u^{2}+C

Replacing u by sec(3x), ∫6 sec^{2}(3x)tan(3x)dx = sec^{2}(3x)+C.