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Maths > Integrals > Integral of the Type e^x[f(x) + f'(x)]dx
Integrals

Integral of the Type e^x[f(x) + f'(x)]dx

In this article, we will see the integration rules to be followed for solving an integral of the type ex [f(x) + f ’(x)], where f ’(x) is the derivative of f(x). We will use integration by parts and some other integration rules to solve these equations.

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Integral of the Type e^x[f(x) + f ‘(x)]dx

To begin with, let’s say
I = ∫ ex [f(x) + f ’(x)] dx

Opening the brackets, we get,
I = ∫ ex f(x) dx + ∫ ex f ’(x) dx = I1 + ∫ ex f ’(x) dx … (1)
Where, I1 = ∫ ex f(x) dx

To solve I1, we will use integration by parts. Let the first function = f1(x) = f(x) and the second function = g1(x) = ex. Therefore,

I1 = f(x) ∫ ex dx – ∫ [df(x)/dx ∫ ex dx] dx
Or, I1 = ex f(x) – ∫ ex f ’(x) dx + C

Substituting the value of I1 in equation (1), we get
I = ex f(x) – ∫ ex f ’(x) dx + ∫ ex f ’(x) dx + C = ex f(x) + C
Thus, ∫ ex [f(x) + f ’(x)] dx = ex f(x) + C … (2)

Let’s look at an example to understand it well.

integration-rules

(Source: blue diamond)

Integration Rules – Example 1

Find ∫ ex {tan–1 x + [1 / (1 + x2)]} dx

Solution: Let, f(x) = tan–1 x.
Therefore, f ’(x) = df(x)/dx = d(tan–1 x)/ dx = 1 / (1 + x2)
Hence, the integrand is of the form: ex [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ ex {tan–1 x + [1 / (1 + x2)]} dx = ex tan–1 x + C

Integration rules – Example 2

Find ∫ [ex (x2 + 1) / (x + 1)2] dx

Solution: We have,
I = ∫ [ex (x2 + 1) / (x + 1)2] dx = ∫ [ex (x2 + 1 + 1 – 1) / (x + 1)2] dx
= ∫ [ex {(x2 – 1) / (x + 1)2] + 2 / (x + 1)2}] dx
Therefore, I = ∫ [ex {(x – 1) / (x + 1)] + 2 / (x + 1)2}] dx

Now, let f(x) = (x – 1) / (x + 1). Therefore,
f ’(x) = df(x) / dx = d [(x – 1) / (x + 1)] / dx = 2 / (x + 1)2
Hence, the integrand is of the form: ex [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ [ex (x2 + 1) / (x + 1)2] dx = ex (x – 1) / (x + 1) + C

More Solved Examples for You

Question 1: Find ∫ ex (sin x + cos x) dx

Answer :f ’(x) = df(x)/dx = d sin x / dx = cos x.
Hence, the integrand is of the form: ex [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ ex (sin x + cos x) dx = ex sin x + C

Question 2: Find ∫ ex [(1 / x) – (1 / x2)] dx

Answer : Let, f(x) = 1/x. Therefore,
f ’(x) = df(x)/dx = d(1/x)/dx = 1/x2.
Hence, the integrand is of the form: ex [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ ex [(1 / x) – (1 / x2)] dx = ex (1/x) + C = ex/x + C

Solved Questions for You

Question 1: What are integration and differentiation?

Answer: Differentiation refers to the act of finding the rate of change of the gradient/slope of any function whereas integration refers to the area under the curve of function with regards to the x-axis.

Question 2: What is the product rule of integration?

Answer: The Product Rule of integration allows us to integrate the product of two functions. For instance, through a series of mathematical somersaults, we can turn any equation into a formula which is useful for integrating.

Question 3: Who introduced integration?

Answer: Gottfried Wilhelm Leibniz formulated the principle of integration along with Isaac Newton.  It originated in the late 17th century where they considered the integral as an infinite sum of rectangles of infinitesimal width.

Question 4: What is called integration?

Answer: Integration basically refers to bringing together and uniting of things. For instance, the integration of two or more economies, cultures, religions and so on. However, in math, integration refers to a concept of calculus, which is the act of finding integrals.

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