Integral of the Type e^x[f(x) + f'(x)]dx: Integration Rules, Solved Examples

In this article, we will see the integration rules to be followed for solving an integral of the type e^x [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 = ∫ e^x [f(x) + f ’(x)] dx

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

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

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

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

integration-rules

(Source: blue diamond)

Integration Rules – Example 1

Find ∫ e^x {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: e^x [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ e^x {tan^–1 x + [1 / (1 + x²)]} dx = e^x tan^–1 x + C

Integration rules – Example 2

Find ∫ [e^x (x² + 1) / (x + 1)²] dx

Solution: We have,
I = ∫ [e^x (x² + 1) / (x + 1)²] dx = ∫ [e^x (x² + 1 + 1 – 1) / (x + 1)²] dx
= ∫ [e^x {(x² – 1) / (x + 1)²] + 2 / (x + 1)²}] dx
Therefore, I = ∫ [e^x {(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)²Hence, the integrand is of the form: e^x [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ [e^x (x² + 1) / (x + 1)²] dx = e^x (x – 1) / (x + 1) + C

More Solved Examples for You

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

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

Question 2: Find ∫ e^x [(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: e^x [f(x) + f ’(x)]. Therefore, using equation (2), we get
∫ e^x [(1 / x) – (1 / x²)] dx = e^x (1/x) + C = e^x/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 17^th 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.