Math Problem Statement
Solution
The image contains two indefinite integrals to solve:
A.
B.
Let's approach them one by one.
A.
For this, we can use u-substitution.
Let , so that .
Now, rewrite the integral:
Now, integrate :
Substitute back for :
B.
Here, we use u-substitution again.
Let , so .
Now, rewrite the integral:
Now, integrate :
Substitute back for :
Final Answers:
- A:
- B:
Would you like more details or help with anything else? Here are some related questions you might find useful:
- What other substitution techniques are useful in integration?
- Can we apply integration by parts to these integrals?
- How can we check the correctness of these indefinite integrals?
- How do we handle definite integrals with the same techniques?
- What are common mistakes in using substitution for integrals?
Tip: Remember to always carefully check the differential in u-substitution. If it's off by a factor, the whole integral will not work correctly!
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Math Problem Analysis
Mathematical Concepts
Integration
U-substitution
Trigonometric Integrals
Formulas
u-substitution formula: ∫f(g(x)) g'(x) dx = ∫f(u) du
Basic integral of cos(u): ∫cos(u) du = sin(u)
Basic integral of uⁿ: ∫uⁿ du = (uⁿ⁺¹)/(n+1)
Theorems
U-substitution
Basic integration techniques
Suitable Grade Level
Grades 11-12 (Calculus)
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