Problems

01

Somewhat odd power product

Compute the integral:

Solution

01

(1) Notice odd power on . Swap the even bunch:

---

(2) Integrate with -sub setting and thus :

Link to original

02

Tangent and secant both even

Compute the integral:

Solution

02

(1) Notice . Therefore integrate with -sub setting and :

Link to original

03

All even power product

Compute the integral:

Solution

03

(1) Notice all even powers. Use power-to-frequency conversion:

Plug in:

Simplify:

---

(2) Reduce power again for :

(This is derived from the power-to-frequency formula by changing ‘’ to ‘’ in that formula.)

---

(3) On the last term, swap even bunch:

Plug all in and obtain:

---

(4) Integrate the first three terms:

---

(5) Integrate the last term with -sub, setting and :

---

(6) Combine in final result:

Note: It is also possible to rewrite using trig identities. So, equally valid answers may look different than this.

Link to original

04

All odd power product

Compute the integral:

Solution

06

(1) Notice odd power on . Swap the even bunch:

---

(2) Perform -sub setting and thus :

---

(3) Convert back to :

Link to original

05

Tangent and secant mixed parity

Compute the integral:

• (a) Using .
• (b) Using .

Solution

07

(a) Select and thus :

---

(b)

(1) Select and thus :

---

(3) Swap even bunch using :

---

(4) Perform -sub with and integrate:

Link to original

06

Power product with negative power

Compute the integral:

Solution

08

(1) Change variable by substituting and :

---

(2) Identify :

---

(3) Perform -sub with and thus :

Link to original