   Chapter 4.5, Problem 6E

Chapter
Section
Textbook Problem

# Evaluate the integral by making the given substitution. ∫ 2 t + 1 d t ,    u = 2 t + 1

To determine

To evaluate:

The integral 2t+1 dt by making the given substitution u=2t+1.

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval 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:

2t+1 dt     u=2t+1

3) Calculation:

Use thesubstitution u=2t+1.

Differentiate u=2t+1 with respect to t.

du=2dt

Divide by 2.

dt=du2

By using concept i),

substitute u=2t+1,dt=du2

=udu2

=u12

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