Definite Integral of cos(x)sin(sin(x)) from 0 to pi/2

TL;DR

Use a u substitution to evaluate the definite integral of cosine times the sine of the sine of x from zero to pi over two.

Transcript

so we have to evaluate the definite integral of cosine times the sine of the sine of x from zero to pi over two so in a problem like this we probably should try to make a u substitution i'm thinking if we let u be sine then the derivative of sine is cosine and so we'll just end up with like the sine of u so that seems to be a good strategy so what'... Read More

Key Insights

• 😄 The u substitution method is a useful technique in evaluating definite integrals.
• 💄 Changing the limits of integration is necessary when making a u substitution.
• 👨‍💼 The integral of sine is negative cosine.
• ⛔ The top limit is always evaluated before the bottom limit in a definite integral.

Questions & Answers

Q: What is the strategy for evaluating the given definite integral?

The strategy is to make a u substitution, letting u be sine, which simplifies the integral to the sine of u.

Q: How does the substitution affect the limits of integration?

When substituting u for sine, the x limits must be converted to u limits. Plugging in the values, u is equal to 0 when x is 0 and u is equal to 1 when x is pi over 2.

Q: What is the integral of sine?

The integral of sine is negative cosine. So, when integrating the sine of u, we get -cosine of u.

Q: How do we determine the value of the definite integral?

Plugging in the top limit, which is 1, we have -cosine of 1. Plugging in the bottom limit, which is 0, we have -cosine of 0. Simplifying, we get 1 minus the cosine of 1.

Summary & Key Takeaways

• To evaluate the definite integral, a u substitution is used.

• Letting u be sine, the integral becomes the sine of u.

• After changing the limits of integration, the integral is simplified to -cosine of 1 plus 1.