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Boost your knowledge of Integration with expert-guided homework solutions.
Integration Homework HelpQ1: What is the integral of \( f(x) = x^2 \) with respect to \( x \)?
See AnswerThe integral of \( x^2 \) with respect to \( x \) is: \[ \int x^2 \, dx = \frac{x^3}{3} + C \] where \( C \) is the constant of integration.
Q2: What is the definite integral \( \int_0^1 (3x^2 + 2x) \, dx \)?
See AnswerThe definite integral \( \int_0^1 (3x^2 + 2x) \, dx \) is: \[ \left[ x^3 + x^2 \right]_0^1 = 1 + 1 - 0 = 2 \]
Q3: Find the integral of \( f(x) = e^x \).
See AnswerThe integral of \( e^x \) with respect to \( x \) is: \[ \int e^x \, dx = e^x + C \]
Q4: What is the integral of \( \sin(x) \) with respect to \( x \)?
See AnswerThe integral of \( \sin(x) \) with respect to \( x \) is: \[ \int \sin(x) \, dx = -\cos(x) + C \]
Q5: Solve the integral \( \int \frac{1}{x} \, dx \).
See AnswerThe integral of \( \frac{1}{x} \) with respect to \( x \) is: \[ \int \frac{1}{x} \, dx = \ln|x| + C \]
Q6: Compute the definite integral \( \int_0^{\pi} \sin(x) \, dx \).
See AnswerThe definite integral \( \int_0^{\pi} \sin(x) \, dx \) is: \[ \left[ -\cos(x) \right]_0^{\pi} = -(-1) - (-1) = 2 \]
Q7: Find the integral of \( \ln(x) \) with respect to \( x \).
See AnswerThe integral of \( \ln(x) \) with respect to \( x \) is: \[ \int \ln(x) \, dx = x \ln(x) - x + C \]
Q8: What is the integral of \( \cos(x) \cdot e^x \)?
See AnswerThis is an integration by parts problem. The solution to \( \int \cos(x) \cdot e^x \, dx \) is: \[ e^x \cdot \sin(x) + C \]
Q9: Evaluate the definite integral \( \int_1^2 \frac{1}{x^2} \, dx \).
See AnswerThe definite integral \( \int_1^2 \frac{1}{x^2} \, dx \) is: \[ \left[ -\frac{1}{x} \right]_1^2 = -\frac{1}{2} + 1 = \frac{1}{2} \]
Q10: Solve the improper integral \( \int_1^{\infty} \frac{1}{x^2} \, dx \).
See AnswerThe improper integral \( \int_1^{\infty} \frac{1}{x^2} \, dx \) is: \[ \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = 1 \]
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