Definite Integrals Practice – Business Calculus Guidance

Q1. Evaluate the definite integral: $\int_{-2}^{3} (4x^3 - 6x + 1)\,dx$

Background

Topic: Definite Integration (Antiderivatives and the Fundamental Theorem of Calculus)

This question tests your ability to compute a definite integral by finding the antiderivative of a polynomial function and evaluating it at the given bounds.

Key Terms and Formulas

• Definite Integral: $\int_{a}^{b} f(x)\,dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

• Antiderivative: A function $F(x)$ such that $F'(x) = f(x)$.

• Fundamental Theorem of Calculus: $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$.

Step-by-Step Guidance

1. Find the antiderivative $F(x)$ of $f(x) = 4x^3 - 6x + 1$. Integrate each term separately:

   • $\int 4x^3\,dx$

   • $\int -6x\,dx$

   • $\int 1\,dx$

2. Combine the results to write the general antiderivative $F(x)$.

3. Apply the Fundamental Theorem of Calculus: $\int_{-2}^{3} f(x)\,dx = F(3) - F(-2)$.

4. Substitute $x=3$ and $x=-2$ into your antiderivative $F(x)$ to set up $F(3)$ and $F(-2)$.

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Q2. Evaluate the definite integral: $\int_{1}^{4} \left(-3x^{-\frac{1}{2}} + x^{\frac{1}{2}}\right)\,dx$

Background

Topic: Definite Integration with Rational Exponents

This question tests your ability to integrate functions with fractional exponents and apply the limits of integration.

Key Terms and Formulas

• Power Rule for Integration: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$.

• Definite Integral: $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

Step-by-Step Guidance

1. Integrate $-3x^{-\frac{1}{2}}$ using the power rule.

2. Integrate $x^{\frac{1}{2}}$ using the power rule.

3. Combine the results to write the antiderivative $F(x)$.

4. Set up $F(4) - F(1)$ using your antiderivative.

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Q3. Evaluate the definite integral: $\int_{1}^{6} (2x^{-1} - e^x)\,dx$

Background

Topic: Definite Integration with Logarithmic and Exponential Functions

This question tests your ability to integrate $x^{-1}$ (which gives a natural logarithm) and exponential functions, then apply the limits of integration.

Key Terms and Formulas

• Integral of $x^{-1}$: $\int x^{-1}\,dx = \ln|x| + C$

• Integral of $e^x$: $\int e^x\,dx = e^x + C$

• Definite Integral: $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$

Step-by-Step Guidance

1. Integrate $2x^{-1}$ to get $2\ln|x|$.

2. Integrate $-e^x$ to get $-e^x$.

3. Combine the results to write the antiderivative $F(x)$.

4. Set up $F(6) - F(1)$ using your antiderivative.

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Q4. Evaluate the definite integral: $\int_{-1}^{2} 3(2t+1)e^{t^2 + t}\,dt$

Background

Topic: Definite Integration Using Substitution (u-substitution)

This question tests your ability to recognize when substitution is needed to integrate a composite function, and then apply the limits of integration.

Key Terms and Formulas

• u-substitution: If $u = g(t)$, then $\int f(g(t))g'(t)\,dt = \int f(u)\,du$

• Definite Integral with Substitution: Change the limits of integration to match the new variable, or substitute back to $t$ before evaluating.

Step-by-Step Guidance

1. Let $u = t^2 + t$. Compute $du$ in terms of $dt$.

2. Express $3(2t+1)dt$ in terms of $du$ to match the integrand.

3. Rewrite the integral in terms of $u$ and adjust the limits if you wish to integrate with respect to $u$.

4. Integrate $e^u$ with respect to $u$ to find the antiderivative.

5. Set up the evaluation of the antiderivative at the new or original limits.

Try solving on your own before revealing the answer!