   Chapter 4.5, Problem 33E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. (take C = 0 ). ∫ sin 3 x cos x   d x

To determine

To evaluate:

The given indefinite integral sin3xcosx dx.

Explanation

1) Concept:

i) The substitution rule:

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then f(gx)g'xdx=f(u)du. ii) Indefinite integral

xn dx=xn+1n+1+C  n-1

2) Given:

sin3xcosx dx

3) Calculation:

The given integral is sin3xcosx dx. Here, use the substitution method.

Substitute sinx=u.

Differentiating with respect to x,

cosx dx=du

Therefore, the given integral becomes

sin3xcosx dx=u3du

Using concept ii),

=u3+13+1+C

=u44+C

Resubstitute sinx=u

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