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Magazine Chapter 7: Problem 8
Find the integrals Check your answers by differentiation. $$\int t \cos \left(t^{2}\right) d t$$
Short Answer
Expert verified
The integral is \( \frac{1}{2} \sin(t^2) + C \).
Step by step solution
01
Identify the method to use
The integral \( \int t \cos(t^2) \, dt \) can be solved using the substitution method. We will choose a substitution that simplifies the integrand.
02
Choose a substitution
Let \( u = t^2 \). Then, the differential \( du = 2t \, dt \), or equivalently \( dt = \frac{1}{2t} \, du \). Notice that \( t \, dt = \frac{1}{2} du \).
03
Substitute and simplify the integral
Substitute \( u = t^2 \) into the integral:\[\int t \cos(t^2) \, dt = \int \cos(u) \frac{1}{2} \, du = \frac{1}{2} \int \cos(u) \, du.\]
04
Integrate the simplified function
The integral \( \int \cos(u) \, du \) is a standard integral:\[\int \cos(u) \, du = \sin(u) + C.\]Thus,\[\frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + C.\]
05
Substitute back to the original variable
Replace \( u \) with \( t^2 \) to get back to the original variable:\[\frac{1}{2} \sin(u) + C = \frac{1}{2} \sin(t^2) + C.\]
06
Differentiate to verify the integral
Differentiate \( \frac{1}{2} \sin(t^2) + C \) with respect to \( t \):\[\frac{d}{dt} \left( \frac{1}{2} \sin(t^2) + C \right) = \frac{1}{2} \cos(t^2) \cdot 2t = t \cos(t^2).\]We obtain the original integrand, confirming that the integration is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a popular integration technique that simplifies the integration of a composite function. It's particularly useful when dealing with integrals where a substitution can turn a complex problem into a more manageable form. When you look at an integral and see a function inside another function, substitution might be the key to solving it.
Here's how to use it effectively:
Here's how to use it effectively:
- Identify a suitable substitution. Look for a function and its derivative within the integral that can reduce complexity.
- Express the original integral in terms of a new variable, say \( u \). Replace all instances of the old variable with \( u \).
- Don't forget to change the differential as well. If \( u = g(t) \), then \( du = g'(t) \, dt \).
- Integrate with respect to \( u \). It's typically simpler than the original function.
- Substitute back into the original variable to get your final answer.
Trigonometric Integrals
Trigonometric integrals involve functions like sine, cosine, tangent, etc. Integrating these requires knowledge of standard trigonometric integrals. These are essentials for solving a range of calculus problems.
- Common trigonometric integrals include \( \int \sin(x) \, dx = -\cos(x) + C \) and \( \int \cos(x) \, dx = \sin(x) + C \).
- The presence of a trigonometric function can often signal the need for substitution. Notice, reversing the process during differentiation often results back to the original trigonometric forms.
Differentiation Verification
Differentiation is a way of confirming the correctness of a solution obtained through integration. It's an essential step that ensures the integrity of your result.
- Once you solve an integral, differentiate the resulting function. The differentiated result should match the original integrand.
- If the differentiated result matches the integrand, the integration was performed correctly. If not, review your steps for possible errors.
- Differentiation is essentially the reverse operation of integration, and thus it provides a natural check on your work.
