9.3.1 Integrating Composite Trigonometric Functions by Substitution

Question 1

Integrating Composite Trigonometric Functions by Substitution

Answer 1

• Integration by substitution is a technique for finding the antiderivative of a composite function. A composite function is a function that results from first applying one function, then another.
• If the du-expression is only off by a constant multiple, you can still use integration by substitution by moving that constant out of the integral.

Question 2

note

Answer 2

• This integral involves a composite function: the sine of a complicated expression. If you let u be the inside of the function, notice that du is found surrounding the sine function.
• After you substitute u, make sure that nothing remains in terms of x.
• Recall that the derivative of –cosx is sinx.
• Make sure to replace u with its expression in terms of x.
• You can check that your answer is correct by taking its
  derivative.
• Here is another composite function. Let u be the inside
  expression. When you find du, you will notice that there is no multiple of 4 in the integrand, just dx.
• Since 4 is just a constant multiple, solve for dx and substitute that expression into the integrand.
• You can move 1/4 outside the integrand since it is a constant multiple.
• After you integrate, make sure to replace u with its expression in terms of x.
• Take the derivative of your answer to make sure it is correct.

Question 3

Solve the integral:∫x^−1/4 csc^2x^3/4dx

Answer 3

−4/3cotx^3/4+C

Question 4

Integrate:∫3x^2sinx^3dx

Answer 4

− cos (x ^3 ) + C

Question 5

Integrate.∫2t(1+t^2)^2sec^2[(1+t^2)^3]dt

Answer 5

1/3tan[(1+t^2)^3]+C

Question 6

Find the integral.∫5xcosx^2dx

Answer 6

5/2sin  x^2+C

Question 7

Evaluate:∫2x^3sinx^4dx.

Answer 7

−1/2cosx^4+C

Question 8

Evaluate the integral.

∫cos√x/√x dx

Answer 8

2sin√x+C

Question 9

Integrate:∫xsec^2(x^2−1)dx.

Answer 9

1/2tan(x^2−1)+C

Question 10

Evaluate:∫√x cscx^3/2 cotx^3/2dx

Answer 10

−2/3cscx^3/2+C

Question 11

Evaluate the integral:∫^3√x⋅sec^2(1−x^4/3)dx

Answer 11

−3/4tan(1−x^4/3)+C

Question 12

Solve the integral.∫(sec2xtan2x) dx

Answer 12

sec2x/2+C