# Problem with solution Simplify 6 Integrals with the Delta Function

Level 3 (with higher mathematics)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

Calculate the following integrals containing a delta function $$\delta(x)$$:

1. Integral 1
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2. Integral 2
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3. Integral 3
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4. Integral 4
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5. Integral 5
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6. Integral 6
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Solution tips
• First, check that the delta function is within the integration limits.
• If the delta function lies within the integration limits, then evaluate $$f(x)$$ at the position $$p$$ of the delta function:$$\int f(x) \, \delta(\cdot) \, \text{d}x ~=~ f(p)$$
Solution for (a)

We want to calculate the following integral:$$\int^5_1 \left( 2x^2 - x + 1 \right) \, \delta(x-3) \, \text{d}x$$Here $$f(x) = 2x^2 - x + 1$$. And the position of the delta function is at $$x=3$$.

First, we ask ourselves if the position of $$\delta(x-3)$$ lies in the integration interval [1, 5]. It does. Therefore, the integral is not necessarily zero and we must evaluate the function $$f(x)$$ at the position $$x=3$$ where the delta function is found:\begin{align} \int^5_1 \left( 2x^2 - x + 1 \right) \, \delta(x-3) \, \text{d}x ~&=~ 2\cdot 3^2 - 3 + 1 \\ ~&=~ 16 \end{align}

Solution for (b)

We want to calculate the following integral:$$\int^{3}_0 x^3 \, \delta(x+2) \, \text{d}x$$Here $$f(x) = x^3$$. And the delta function is shifted to the negative and is at $$x=-2$$.

First, we ask ourselves if the position of $$\delta(x+2)$$ lies in the integration interval [0, 3]. It does not. The integral is zero:\begin{align} \int^{3}_0 x^3 \, \delta(x+2) \, \text{d}x ~=~ 0 \end{align}

Solution for (c)

We want to calculate the following integral:$$\int^4_0 \cos(x) \, \delta(x-\pi) \, \text{d}x$$Here $$f(x) = \cos(x)$$. And the position of the delta function is at $$x=\pi$$.

First, we ask ourselves if the position of $$\delta(x-\pi)$$ lies in the integration interval [0, 4]. It does. Therefore, the integral is not necessarily zero and we must evaluate the function $$f(x)$$ at the position $$x=\pi$$ where the delta function is found:\begin{align} \int^4_0 \cos(x) \, \delta(x-\pi) \, \text{d}x ~&=~ \cos(\pi) \\ ~&=~ -1 \end{align}

Solution for (d)

We want to calculate the following integral:$$\int \ln(x+3) \, \delta(x+1) \, \text{d}x$$Here $$f(x) = \ln(x+3)$$. And the position of the delta function is at $$x=-1$$.

First, we ask ourselves if the position of $$\delta(x+1)$$ lies in the integration interval $$[-\infty, \infty]$$. It does. Therefore, the integral is not necessarily zero and we must evaluate the function $$f(x)$$ at the position $$x=-1$$ where the delta function can be found:\begin{align} \int \ln(x+3) \, \delta(x+1) \, \text{d}x ~&=~ \ln(-1 + 3) \\ ~&=~ 0.693... \end{align}

Solution for (e)

We want to calculate the following integral:$$\int^2_{-3} \left( 6x + 2 \right) \, \delta(3x) \, \text{d}x$$Here $$f(x) = \left( 6x + 2 \right)$$. And the position of the delta function is at $$x=0$$. The scaling factor is $$|k| = 3$$.

First we question whether the position of $$\delta(3x)$$ lies in the integration interval $$[-3, 2]$$. It does. Therefore, the integral is not necessarily zero and we need to evaluate the function $$f(x)$$ at the position $$x=0$$ and multiply it by the factor $$1/|k|$$:\begin{align} \int^2_{-3} \left( 6x + 2 \right) \, \delta(3x) \, \text{d}x ~&=~ \frac{1}{3}\, \left( 6\cdot 0 + 2 \right) \\ ~&=~ \frac{2}{3} \end{align}

Solution for (f)

We want to calculate the following integral:$$\int^b_{-\infty} 3 \, \delta(x-a) \, \text{d}x$$Here $$f(x) = 3$$ is a constant function. And the position of the delta function is at $$x=a$$.

First we question whether the position of $$\delta(x-a)$$ is in the integration interval $$[-\infty, b]$$. This depends on whether $$a$$ is greater or smaller than $$b$$.

• If $$a$$ is GREATER than $$b$$, then the delta function is outside the integration limits and the integral is zero in this case.
• If $$a$$ is SMALLER than $$b$$, then the delta function is within the integration limits and the integral is not necessarily zero in this case. Here we have to evaluate the function $$f(x)$$ at the point $$x=a$$:$$f(a) = 3$$

The overall result is thus:$$\int^b_{-\infty} 3 \, \delta(x-a) \, \text{d}x \begin{cases} 3, &\mbox{} a < b \\ 0, &\mbox{} a > b \end{cases}$$