Question

Evaluate the following integrals or state that they diverge. $$\int_{0}^{10} \frac{d x}{\sqrt[4]{10-x}}$$

Short answer

Question: Evaluate the given integral: $$\int_{0}^{10} \frac{d x}{\sqrt[4]{10-x}}$$ Answer: The value of the integral is: $$\frac{10^{\frac{3}{4}}}{\frac{3}{4}}$$

Step-by-step solution

Substitution

Let \(u=10-x\), then \(d u=-dx\). Our bounds also change accordingly. When \(x=0, u=10\) and when \(x=10, u=0\). So, our integral now becomes: $$-\int_{10}^{0} \frac{d u}{u^{\frac{1}{4}}}$$

Change the order of integration

We can switch the bounds by changing the sign of the integral: $$\int_{0}^{10} \frac{d u}{u^{\frac{1}{4}}}$$

Integrate the function

To integrate the function \(\frac{1}{u^{\frac{1}{4}}}\), we can rewrite it as \(u^{-\frac{1}{4}}\) and use the power rule of integration: $$\int u^{-\frac{1}{4}} d u = \frac{u^{(-\frac{1}{4}+1)}}{(-\frac{1}{4}+1)} + C = \frac{u^{\frac{3}{4}}}{\frac{3}{4}} + C$$ Now let's evaluate the integral with the new bounds:

Evaluate the integral with the new bounds

Substitute the new bounds (0 and 10) into the antiderivative and compute the result: $$\frac{10^{\frac{3}{4}}}{\frac{3}{4}} - \frac{0^{\frac{3}{4}}}{\frac{3}{4}} = \frac{10^{\frac{3}{4}}}{\frac{3}{4}}$$

Convert back to original variable

Now let's convert back to the original variable using the substitution \(u = 10-x\): $$\frac{(10-x)^{\frac{3}{4}}}{\frac{3}{4}}\Big|_{0}^{10} = \frac{10^{\frac{3}{4}}}{\frac{3}{4}}-\frac{0^{\frac{3}{4}}}{\frac{3}{4}} = \frac{10^{\frac{3}{4}}}{\frac{3}{4}}$$

Final Answer

The value of the integral is: $$\int_{0}^{10} \frac{d x}{\sqrt[4]{10-x}} = \frac{10^{\frac{3}{4}}}{\frac{3}{4}}$$

Key concepts

Integration Techniques

Integration is a fundamental part of calculus, used for finding areas, volumes, central points, and many more useful things. Different integration techniques are required to tackle various kinds of functions.

Some common integration techniques include the substitution method, integration by parts, partial fractions, and trigonometric integrals.

Each technique has its place, and no single method can solve every integral. For instance, the substitution method, used in the provided exercise, is especially powerful when you notice a part of the integrand can be replaced by a single variable to simplify the integral.

Substitution Method

The substitution method is a technique often employed when an integral involves a composition of functions. It works similarly to finding the antiderivative of a chain rule derivative in reverse.

In essence, you choose a part of the integrand to replace with a new variable, making the integral easier to manage. This process involves finding a suitable substitution that simplifies the integrand and changing the limits of integration if dealing with a definite integral, just like in the step-by-step solution we see with the substitution of u = 10 - x.

Definite Integrals

Probably the most practical use of integration is the definite integral, which is used to calculate the area under a curve within set bounds.

To solve a definite integral, you first find the indefinite integral, which is the general antiderivative of the function. After that, you evaluate this antiderivative at the upper and lower bounds of the integral and subtract these values. This process is illustrated in the exercise when we evaluate the antiderivative from 0 to 10.

Antiderivatives

An antiderivative of a function, often called an indefinite integral, represents all the different functions that could be differentiated to get the original function.

Antiderivatives are not unique because the differentiation of a constant is zero; hence a family of functions differs by a constant can have the same derivative. This is why we add 'C', the constant of integration, when we find the antiderivative. In definite integrals, 'C' cancels out, so it is not usually written out as part of the steps, similar to the provided solution.

Power Rule of Integration

One of the most commonly used techniques in integration is the power rule, which states that for any real number n different from -1, the antiderivative of x^n with respect to x is (x^(n+1))/(n+1) plus the constant of integration, C.

This rule is essential in our exercise, as we apply it to integrate u^(-1/4). By adding 1 to the exponent and dividing by the new exponent, we find the antiderivative, which is integral in calculating definite integrals of polynomials or any function that can be simplified to a power of x.