Is My Integral Evaluation Correct?

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In summary, the process for evaluating integrals involves simplifying the integral using algebraic manipulations, choosing an appropriate integration technique (such as substitution, integration by parts, or partial fractions), and substituting in the limits of integration to find the exact value. The choice of technique depends on the form of the integral. Integrals can be evaluated using a calculator, but this may only provide an approximation. The constant of integration is important as it represents all possible antiderivatives and must be included when solving indefinite integrals. Common mistakes to avoid include forgetting the constant of integration, misapplying techniques, and making errors in algebraic simplification. It is also important to check for potential discontinuities or singularities in the integrand
  • #1
shamieh
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Evaluate the Integral.

Just need someone to check my work.
\(\displaystyle
\int sec^4y \, tan^4y\)

\(\displaystyle \tan^4y * sec^2y * sec^2y \, dx\)

\(\displaystyle tan^4y * (1 + tan^2y) * sec^2y\)

\(\displaystyle u = tany\)
\(\displaystyle du = sec^2y\)

\(\displaystyle \int u^4 * (1 + u^2) * du\)

\(\displaystyle \int u^4 + u^6 * du\)

\(\displaystyle \frac{u^5}{5} + \frac{u^7}{7} + C\)

\(\displaystyle \frac{tan^5x}{5} + \frac{tan^7x}{7} + C\)
 
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  • #2
Looks good to me :D
 

FAQ: Is My Integral Evaluation Correct?

What is the process for evaluating integrals?

The process for evaluating integrals involves using techniques such as substitution, integration by parts, and partial fractions. First, the integral needs to be simplified using algebraic manipulations. Then, one of the integration techniques is applied to find the antiderivative. Finally, the limits of integration are substituted into the antiderivative to find the exact value of the integral.

How do I know which integration technique to use?

The choice of integration technique depends on the form of the integral. For example, substitution is used when the integrand contains a function and its derivative, integration by parts is used when the integrand contains a product of functions, and partial fractions is used when the integrand contains a rational function. It is important to recognize the form of the integral in order to choose the appropriate technique.

Can integrals be evaluated using a calculator?

Yes, some calculators have the capability to evaluate integrals numerically. However, this method will only provide an approximation of the integral and may not be accurate for more complex integrals. It is still important to understand the process of evaluating integrals by hand.

What is the significance of the constant of integration?

The constant of integration is added to the antiderivative when evaluating integrals. This constant is important because it represents all possible antiderivatives of the original function. When solving a definite integral, the constant is cancelled out, but it is necessary to include it when finding an indefinite integral.

Are there any common mistakes to avoid when evaluating integrals?

One common mistake is forgetting to include the constant of integration. Another mistake is misapplying the integration techniques or making errors in algebraic simplification. It is important to double check your work and be careful with signs and exponents. Also, make sure to check for any potential discontinuities or singularities in the integrand that may require special consideration.

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