Integrate Using u-Substitution integral from 1 to 3 of (x^3-4x) with respect to x

The question asks for the computation of a definite integral using the method of u-substitution. The integral to be evaluated is the integral of (x^3 - 4x) with respect to x, ranging from the lower limit of 1 to the upper limit of 3. U-substitution is a technique used in calculus to simplify the integration process where a part of the integrand (the expression within the integral sign) is substituted with a new variable, typically denoted as 'u'. This requires identifying a portion of the integrand that can be represented as 'u', then substituting it and its differential 'du' into the integral. Once the substitution is made, the integral is solved in terms of 'u', and then the variable 'u' is replaced with the original terms to get the answer in terms of the original variable x. The final step is to evaluate the resulting expression at the upper and lower limits of the integral. The question does not require the actual execution of these steps but an understanding of the process and what is being asked.

$\int_{1}^{3} \left(\right. x^{3} - 4 x \left.\right) d x$

Answer

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Solution:

Step:1

We cannot apply u-substitution to this integral. We will proceed with a different technique.

Step:2

First, we expand the integral expression: $\int_{1}^{3} x^{3} - 4x \, dx$

Step:3

Separate the integral into two parts: $\int_{1}^{3} x^{3} \, dx - \int_{1}^{3} 4x \, dx$

Step:4

Apply the Power Rule to integrate $x^{3}$: $\frac{1}{4} x^{4} \Big|_{1}^{3} - \int_{1}^{3} 4x \, dx$

Step:5

Extract the constant $-4$ from the integral: $\frac{1}{4} x^{4} \Big|_{1}^{3} - 4 \int_{1}^{3} x \, dx$

Step:6

Integrate $x$ using the Power Rule: $\frac{1}{4} x^{4} \Big|_{1}^{3} - 4 \left( \frac{1}{2} x^{2} \Big|_{1}^{3} \right)$

Step:7

Now, we simplify the expression.

Step:7.1

Combine the constants with the powers of $x$: $\frac{1}{4} x^{4} \Big|_{1}^{3} - 4 \left( \frac{x^{2}}{2} \Big|_{1}^{3} \right)$

Step:7.2

Substitute the limits of integration and simplify.

Step:7.2.1

Evaluate $\frac{1}{4} x^{4}$ at the upper and lower limits: $\left( \frac{1}{4} \cdot 3^{4} \right) - \left( \frac{1}{4} \cdot 1^{4} \right) - 4 \left( \frac{x^{2}}{2} \Big|_{1}^{3} \right)$

Step:7.2.2

Evaluate $\frac{x^{2}}{2}$ at the upper and lower limits: $\frac{1}{4} \cdot 3^{4} - \frac{1}{4} \cdot 1^{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3

Proceed with simplification.

Step:7.2.3.1

Calculate $3^{4}$: $\frac{1}{4} \cdot 81 - \frac{1}{4} \cdot 1^{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.2

Combine the constant $\frac{1}{4}$ with $81$: $\frac{81}{4} - \frac{1}{4} \cdot 1^{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.3

Any number to the power of one is itself: $\frac{81}{4} - \frac{1}{4} \cdot 1 - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.4

Multiply $-1$ by $1$: $\frac{81}{4} - \frac{1}{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.5

Combine the numerators over a common denominator: $\frac{81 - 1}{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.6

Subtract $1$ from $81$: $\frac{80}{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.7

Reduce the fraction by canceling common factors.

Step:7.2.3.7.1

Factor out $4$ from $80$: $\frac{4 \cdot 20}{4} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.7.2

Cancel out the common factors.

Step:7.2.3.7.2.1

Factor $4$ out of $4$: $\frac{4 \cdot 20}{4 \cdot 1} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.7.2.2

Cancel the common factor: $\frac{\cancel{4} \cdot 20}{\cancel{4} \cdot 1} - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.7.2.3

Rewrite the expression: $20 - 4 \left( \frac{3^{2}}{2} - \frac{1^{2}}{2} \right)$

Step:7.2.3.7.2.4

Divide $20$ by $1$: $20 - 4 \left( \frac{9}{2} - \frac{1}{2} \right)$

Step:7.2.3.8

Calculate $3^{2}$: $20 - 4 \left( \frac{9}{2} - \frac{1}{2} \right)$

Step:7.2.3.9

Any number to the power of one is itself: $20 - 4 \left( \frac{9}{2} - \frac{1}{2} \right)$

Step:7.2.3.10

Combine the numerators over a common denominator: $20 - 4 \cdot \frac{9 - 1}{2}$

Step:7.2.3.11

Subtract $1$ from $9$: $20 - 4 \cdot \frac{8}{2}$

Step:7.2.3.12