   Chapter 4.5, Problem 32E

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 ). ∫ tan 2 θ sec 2 θ   d θ

To determine

To evaluate:

The given indefinite integral tan2θsec2θdθ

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:

tan2θsec2θdθ

3) Calculation:

The given integral is tan2θsec2θdθ. Here, use the substitution method.

Substitute tanθ=u.

Differentiation with respect to θ.

sec2θdθ=du

Therefore, the given integral becomes

tan2θsec2θdθ=u2du

Using concept ii),

=u2+12+1+C

=u33+C

=13u3+C

Resubstitute  tanθ=u

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