Question

Find the indefinite integral using the substitution x=2sin(θ). (Remember to use absolute values where appropriate. Use C for the constant of integration ∫x4−x2​​dx

Answer 1

`\[ (1)/(3)x^3 - (1)/(5)x^5 + C \]`

Step-by-step explanation:

To find the indefinite integral of
`\( \int (x^4 - x^2) \,dx \)` using the substitution
`\( x = 2\sin(\theta) \), we first need to express \( dx \) in terms of \( d\theta \).`Differentiating both sides of
`\( x = 2\sin(\theta) \) gives \( dx = 2\cos(\theta) \,d\theta \).`

Substituting this into the integral, we get
`\( \int (x^4 - x^2) \,dx = \int (16\sin^4(\theta) - 4\sin^2(\theta)) \cdot 2\cos(\theta) \,d\theta \).`

Simplifying further, we have
`\( 32\int (\sin^4(\theta) - 2\sin^2(\theta))\cos(\theta) \,d\theta \).` Now, we can integrate each term separately. The integral of
`\( \sin^4(\theta) \) involves powers of \( \sin^2(\theta) \),` and the integral of
`\( \sin^2(\theta) \) involves \( \sin(\theta) \)`. After integrating, we substitute back
`\( x = 2\sin(\theta) \)`to obtain the final answer:

`\[ (1)/(3)x^3 - (1)/(5)x^5 + C \]`

where
`\( C \)`is the constant of integration.

Answer 2

To solve the integral ∫ x^4 - x^2 dx with the substitution x=2sin(θ), we find dx in terms of θ, express x's powers with sin(θ), perform the integration with respect to θ, and translate the result back into terms of x, including the constant of integration C.

Step-by-step explanation:

To solve the indefinite integral ∫ x^4 - x^2 ​dx using the substitution x = 2sin(θ), we first need to find the corresponding differential dx and also express the powers of x in terms of sin(θ). Since x = 2sin(θ), we differentiate both sides with respect to θ to find dx. This gives us dx = 2cos(θ)dθ. Substituting x values we get:

x^2 = (2sin(θ))^2 = 4sin^2(θ)

x^4 = (2sin(θ))^4 = 16sin^4(θ)

Now we can rewrite the integral in terms of θ:

∫(16sin^4(θ) - 4sin^2(θ)) · 2cos(θ)dθ

After simplifying and integrating with respect to θ, we get an expression in terms of θ, which we then rewrite back in terms of x using inverse trigonometrical functions and the original substitution. Finally, we add the constant of integration C to complete the indefinite integral.