Integrating the product of the power of secant by the power of tangent

Method

We consider the following cases:

1) n even let u = tanx and use the following identity: sec²x = tan²x + 1
2) m odd  u = secx  tan²x = sec²x -1
2) m even n odd reduce power of secx: tan²x = sec²x -1


Example I

Calculate ∫tan²xsec⁴xdx

sec⁴x is the near derivative of tan²x and n is even we write u = tanx.

Taking the derivative of both sides we have du = sec²xdx. dx = du/sec²x.

Let's substitute tanx and dx

 ∫tan²xsec⁴xdx = ∫u²sec⁴x.du/sec²x

                     =  ∫u²sec²xdu

We use the identity sec²x = tan²x + 1 in order to have the integral as a function of u. Then by substituting tanx by u: sec²x = u² + 1. Therefore:

 ∫tan²xsec⁴xdx =  ∫u²( u² + 1)du
                     
                       =  ∫(u⁴ + u²)du
                   
                      = u⁵/5 + u³ + C
                   
                      = 1/5tan⁵x + 1/3tan³x + C

Example II

Solve ∫tan³xsec³xdx.

We start by looking at the condition for n. Here n is odd. There isn't a condition for n only. We look at the condition for m. Here m is odd then we write u = secx and tan²x = sec²x-1

Let's write u = secx. Then du = tanxsecxdx. Therefore dx = du/tanxsecx.

Let's substitute secx and dx.


 ∫tan³xsec³xdx = ∫tan³xu³.du/tanxu
                   
                      =  ∫tan²xu²du
Let's express tanx in function of secx so that we can have the integral as a function of u

∫tan³xsec³xdx =  ∫(sec²x - 1)u²du
                   
                     =  ∫(u² - 1)u²du

                    =   ∫u⁴du -  ∫u²du
       
                   = 1/5u⁵ - 1/3u³ + C
               
                 =  1/5sec⁵x -  1/3sec³x + C

Practice

Solve:

1) ∫tan²xsec³x
2) ∫tan³xsec⁴x

Interested in learning more about integrals visit Center for Integral Development
Intrested in private tutoring visit New Direction Education Services for contact information.

Integrating the power of secants and tangents

In this post you are going to learn how to integrate the power of secant and tangent. There are some formulas that allow to calculate these powers. Before to learn these formulas let's learn first the formulas to integrate the secant and tangent.

Formulas to integrate the function tangent and secant

You learn first the expression of tangent since if you know its integral it is easy to know the integral of secant. The integral of each of these functions has the function ln, sec.

The integral of tangent is an expression of secant

∫tanx = ln❘secx❘ + C

The integral of secant is an expression of sec and tangent. We just add tangent between the absolute bars.from the integral of tanx

∫secx = ln❘secx + tanx❘ + C

N.B. These formulas can be demonstrated but the demonstration is not done here.

Formulas to integrate the power of tangent and secant

Integral of the power of secant

The integral of the power of secant is made of two terms. The first term is a quotient. The second term is the product of a constant by the integral of secant. We are going to learn the formula according to the following steps:

1. The numerator of the quotient and the and the second part of the integral are the same. It is obtained by decreasing the exponent of secx by 2.

∫secⁿ x = secⁿ⁻²x +  ∫secⁿ⁻²x

2. The denominator of the first quotient is obtained by taking only the exponent n and decreasing it by 1

∫secⁿ x = secⁿ⁻²x/n-1 +  ∫secⁿ⁻²x

3. In the first term we take the quotient of the exponent (n-2) of secx and the denominator (n-1). We then obtain the quotient before the integral

∫secⁿ x = secⁿ⁻²x/n-1 + n-2/n-1  ∫secⁿ⁻²x. This is the formula of the integral of the power of secant. We just learn how to memorize  the formula.

Example. Calculate the integral  ∫sec³x

We apply the formula ∫secⁿ x = secⁿ⁻²x/n-1 + n-2/n-1  ∫secⁿ⁻²x.

We substitute n by 3:

  ∫sec³x =  sec³⁻²x/3-1 + 3-2/3-1  ∫sec³⁻²x.

            =   secx/2 + 1/2 ∫secx
         
            =  1/2secx +ln ❘secx + tanx❘ + C

Integral of the power of tangent

The integral of the power of tangent is given by the formula: ∫ tanⁿxdx = tanⁿ⁻ⁱx/n-1- ∫tanⁿ⁻²xdx

 Example. Solve ∫tan⁵xdx

∫tan⁵xdx = tan⁴x/4 - ∫ tan³xdx

 Let's calculate ∫ tan³xdx:

∫ tan³xdx = tan²x/2 - ∫ tanxdx
                                 
              =    tan²x/2 - ln|secx| + C
              =    tan²x/2 + ln|cosx| + C

Let's substitute  tan³xdx:

∫tan⁵xdx = tan⁴x/4 - tan²x/2 - ln|cosx| + C

Practice

Solve:

1) ∫ sec⁵xdx
2)  ∫tan³xdx

Interested in learning more about Calculus visit Center for Integral Development
For private tutoring visit New Direction Education Services