How Do You Solve the Integral of sin^3(x)cos^6(x)dx?

In summary, a trigonometric integral is an integral that involves trigonometric functions and is used to find the area under a curve. To solve a trigonometric integral, various techniques such as substitution and integration by parts can be used. A calculator can also be used, but understanding the concepts is important for accuracy. Some common trigonometric identities used in solving integrals include Pythagorean, double angle, and half-angle identities. Trigonometric integrals are important in fields such as physics, engineering, and calculus, as they help solve real-world problems and understand periodic functions.
  • #1
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Homework Statement


∫sin^3xcos^63xdx


Homework Equations





The Attempt at a Solution


∫sin^3xcos^63xdxsin^3xcos^63xdx

let u=cosx
du=-sinxdx
-du=sinxdx

=∫sin^2xcos^63xsinxdx
=∫(1-cos^2x)cos^63x-du
=-∫(1-u^2)u^63du
=-∫(u^63-u^65)du
=-u^64/64-u^66/66+c
=(-cos^64x/64)-(cos^66x/66)+c

^ I'm thinking this is not the correct answer?
 
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  • #2
Try differentiating it and see if you recover the integrand.
 

FAQ: How Do You Solve the Integral of sin^3(x)cos^6(x)dx?

What is a trigonometric integral?

A trigonometric integral is an integral that involves trigonometric functions, such as sine, cosine, tangent, etc. It is used to find the area under a curve that involves these functions.

How do I solve a trigonometric integral?

To solve a trigonometric integral, you can use various integration techniques such as substitution, integration by parts, or trigonometric identities. It is important to carefully choose the appropriate method for each integral to get the correct solution.

Can I use a calculator to solve a trigonometric integral?

Yes, you can use a calculator to solve a trigonometric integral. However, it is important to understand the concepts and techniques behind solving these integrals to verify the accuracy of the calculator's results.

What are some common trigonometric identities used in solving integrals?

Some common trigonometric identities used in solving integrals include the Pythagorean identities, double angle identities, and half-angle identities. These identities can help simplify the integral and make it easier to solve.

Why are trigonometric integrals important?

Trigonometric integrals are important in various fields of science and mathematics, such as physics, engineering, and calculus. They are used to solve real-world problems involving periodic functions and can also help in understanding and graphing these functions.

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