Integration by Substitution

Introduction

Integration by substitution is a standard integration approach used when the function to be integrated is either a complex function or when direct integration is not possible. Integration by substitution simplifies the integral of a function by converting the provided function to a more straightforward function.

Let’s look at how to integrate by substitutions, as well as some of the most essential substitutions and solved examples.

Method of integration by substitution: A step-by-step guide

The instructions below will help you in completing this approach of integration by substitution.

Step 1: For the given function to be reduced, choose a new variable t.

Step 2: Where f(x) is integrated with respect to x, find the value of dx for the given integral.

Step 3: Make the necessary changes to the function f(x), as well as the new value dx.

Step 4: Integrate the function that you got from the substitution.

Step 5: To get the final solution, replace the original variable x in the final answer.

Here is how it is done,

To find I=∫f(x)dx

Take x=g(t), after differentiating dxdt=g'(t)dx=g'(t)dt

Substituting values I=∫f(x)dx=∫f(g(t))g'(t)dt

Let us help you understand this in another way,

The chain rule for derivatives corresponds to substitution for integrals.

Take F(u)as antiderivative of f(u):

∫f(u)du=F(u)+c

Take u=u(x)is a differentiable function, and the chain rule is applied

ddx∫F(u(x))=F'(u(x))u'(x)=f(u(x))u'(x)

Integrating both sides

∫f(u(x))u'(x)dx=F(u(x))+c

∫f(u(x))u'(x)dx=∫f(u)du, where u=u(x)

When making a substitution for a function whose derivative is also included in the integer, the integration by substitution approach comes in handy. The function becomes more straightforward as a result, and the basic integration formula can be used to integrate it.

When the supplied function f(x) is multiplied by the derivative of the given function f(x)’, i.e. of the form g(f(x)) f(x)’ dx, the integrations by substitution method can be utilized. When the function to be integrated is not in a standard form, a proper substitution can occasionally change it into an integrable form.

Substitution does not have a set formula. A careful examination of the integrand’s shape will aid in the selection of the function for which substitution is to be made. However, as in the example before, one must ensure that the derivative of the function chosen is provided along with dx. A simple adjustment of a constant may be required on occasion. Any variable symbol, such as s, t, u, v, w, x, y, and z, can be substituted for the variable of the provided integral. The original variable should be restored after the integration is complete.

Important substitutions in integrations by substitution

The following are some of the most valuable substitutes for simplifying the provided statement and making the integration process go more smoothly. Let’s take a look at the precise substitutions below for the methods of integration by substitution.

• We use x=asin or x=acos for the integral function f(a^2-x^2)
• We use x=atan or x=acot for the integral function f(x^2+a^2)f(x^2+a^2)
• We use x=acos2 for the integral functions fa-xa+x and fa+xa-x

Some important integrals of trigonometric functions: substitution method

• ∫tan x dx=logsecx+c

We know that tanx=sinxcosx,

Integrating, ∫tan x dx=∫sinxcosxdx

Substituting, cosx=t, then, sin x dx= -dt.

Now, ∫tan x dx= -∫dtt= -logcosx+c

Or, ∫tanxdx=logsecx+c

• ∫cotxdx=logsinx+c

We know that cotx=cosxsinx,

Integrating, ∫cotxdx=∫cosxsinxdx

Substituting, sinx=t, then, cosxdx= -dt.

Now ∫cotxdx=∫dtt=logt+c=logsinx+c

• ∫secxdx=logsecx+tanx+c

Multiply and divide ∫secxdx by (secx+tanx)

∫secxdx=∫{secx(secx+tanx)dx}/(secx+tanx)

Substituting (secx+tanx)=t, secx(secx+tanx)dx=dt

Putting values∫secxdx=∫dtt=logt+c=logsecx+tanx+c

• ∫cosecxdx=logcosecx-cotx+c

Multiply and divide ∫cosecxdx by (cosecx-cotx)

 ∫cosecxdx= ∫{cosecx(cosecx-cotx)dx}/(cosecx-cotx)

Substituting cosecx-cotx=t-cosecx(cosecx-cotx)dx=dt

Putting values ∫secxdx=∫dt/t=logt+c=logcosecx-cotx+c

Let us take a look at some questions:

Let’s find the integral of etan^-1x+x^2

Expression is: ∫etan^-1x+x^2dx

Take tan^-1x=t and differentiate it: d/dxtan^-1x=dt/dx

11+x^2=dt/dx

dx=(1+x^2)dt

Substitute the value:

∫e^tan^-1x+x^2dx=∫e^t(1+x^2)1+x^2dt=∫e^tdt=e^t+c=e^tan^-1x+c

Let’s find the integral of 2xsec^2(x^2+1)

Expression is: ∫2xsec^2(x^2+1)dx

Take x^2+1=t and differentiate it: d/dx(x^2+1)=dt/dx

2x=dt/dx=dt/2x

Substitute the value:

∫2xsec^2(x^2+1)dx=∫2xsec^2(x^2+1)dt2x=∫sec^2tdt=tant+c=tan(x^2+1)+c

Conclusion

We learned about the method of integration by substitution and why it is used. We also discussed the step-by-step process of calculating the integral by substitution process. We looked at some important integrals of trigonometric functions using substitution.