#### To determine

**To evaluate:**

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

#### Answer

13(2t+1)32+C

#### 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

=∫u12du2

By using concept ii),

=12u12+112+1+C

=12u3232+C

=1(2)u3232+C

Cancelling out common factor:

=u323+C

Rearranging,

=u323+C

Resubstitute u=2t+1.

=13(2t+1)32+C

**Conclusion:**

Therefore,

∫2t+1 dt=13(2t+1)32+C