Integrals Involving Trigonometric Functions

What is a common method used to simplify integrals involving trigonometric functions?

• Substitution

• Integration by parts

• Partial fraction decomposition

• Trigonometric substitution

In the integral \(\int{\cos x\sin ^5 x\,dx}\), what substitution is used to simplify the integral?

• \(u = \sin x\)

• \(u = \cos x\)

• \(u = x\)

• \(u = \tan x\)

What is the integral \(\int{{{{\sin }^5}x\,dx}}\) rewritten as using trigonometric identities?

• \(\int{{{{\left( {1 - {{\cos }^2}x} \right)}^2}\,\sin x\,dx}}\)

• \(\int{{\cos^5 x\,dx}}\)

• \(\int{{\sin^4 x\,dx}}\)

• \(\int{{\sin^2 x\,dx}}\)

When dealing with \(\int{{\sin^6 x \cos^3 x\,dx}}\), what is the strategy used based on the parity of the exponents?

• Extract one cosine function and convert the rest to sines

• Extract one sine function and convert the rest to cosines

• Use half-angle formulas

• Use double-angle formulas

What advanced techniques are required when both exponents in the product of sine and cosine functions are even?

• Half-angle and double-angle formulas

• Substitution

• Integration by parts

• Partial fraction decomposition