Chapter 5: 1P (page 242)

Verify each of the following answers for an indefinite integral by one or more of the methods suggestedabove.
1. $\int 2 \sin θ \cos θ d θ = \sin^{2} θ$ or $- \cos^{2} θ$ or $- \frac{1}{2} \cos 2 θ$ Hint: Use trig identities.

Short Answer

Expert verified

It is proved that $\int 2 \sin θ \cos θ d θ = \sin^{2} θ$ .

Step by step solution

The substitution method

The $u -$ substitution reserves the chain rule and it is used to solve integrals.

Itis a methodto obtain antiderivatives. Generally, formula for $u -$ substitution is:

$\int f (g (x)) g' (x) d x = \int f (u) d u$

Here, $u = g (x); d u = g' (x) d x$ .

Evaluate the trigonometric identity

The given trigonometric integral is $\int 2 \sin θ \cos θ d θ$ .

Take the constant out by using the property $\int a . f (x) d x = a . \int f (x) d x$ .

So, after using the above formula in the trigonometric identity, we get:

$\int 2 \sin θ \cos θ d θ = 2 . \int \sin (θ) \cos (θ) d (θ)$

Applying substitution

Let $u = \sin θ$ , then $d u = \cos θ d θ$ .

After applying u- substitution method, we get:

role="math" localid="1653045028107" $2 . \int \sin θ \cos θ d θ = 2 . \int u d u = = 2 . \frac{u^{2}}{2} [Applying Power Rule]$

Now, substituting back $u = \sin θ$ , we get:

$u = \sin^{2} θ$

Simplification of the identity

On further simplification:

$2 . \frac{\sin^{2} θ}{2} = \sin^{2} θ$

Hence, $\int 2 \sin θ \cos θ d θ = \sin^{2} θ$ .

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Verify each of the following answers for an indefinite integral by one or more of the methods suggestedabove.
1. $\int 2 \sin θ \cos θ d θ = \sin^{2} θ$ or $- \cos^{2} θ$ or $- \frac{1}{2} \cos 2 θ$ Hint: Use trig identities.

Short Answer

Step by step solution

The substitution method

Evaluate the trigonometric identity

Applying substitution

Simplification of the identity

One App. One Place for Learning.

Most popular questions from this chapter

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Verify each of the following answers for an indefinite integral by one or more of the methods suggestedabove.1. ∫2sinθcosθdθ=sin2θor -cos2θor -12cos2θ Hint: Use trig identities.

Short Answer

Step by step solution

The substitution method

Evaluate the trigonometric identity

Applying substitution

Simplification of the identity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Electrostatics

Medical Physics

Engineering Physics

Kinematics Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Verify each of the following answers for an indefinite integral by one or more of the methods suggestedabove.
1. $\int 2 \sin θ \cos θ d θ = \sin^{2} θ$ or $- \cos^{2} θ$ or $- \frac{1}{2} \cos 2 θ$ Hint: Use trig identities.