IB Mathematics SL 5.10 Indefinite integral of xn , sinx, cosx, and ex AA SL Paper 1- Exam Style Questions- New Syllabus
Question
Most-appropriate topic codes (Mathematics: analysis and approaches guide):
• SL 5.10: Indefinite integral of \( \sin x, \cos x, e^x \); composites with the linear function \( ax+b \)— Step 1
▶️ Answer/Explanation
Step 1 — Integrate \( g'(x) \):
To find \( g(x) \), integrate the derivative function:
\( g(x) = \int (\cos x + e^{2x}) \, dx \)
Using the standard integrals and the reverse chain rule for \( e^{2x} \):
\( g(x) = \sin x + \frac{1}{2} e^{2x} + C \)
where \( C \) is the constant of integration.
Step 2 — Use initial condition \( g(0) = 7 \):
Substitute \( x = 0 \) and set the function value to 7:
\( g(0) = \sin(0) + \frac{1}{2} e^{2(0)} + C = 7 \)
\( 0 + \frac{1}{2}(1) + C = 7 \)
\( \frac{1}{2} + C = 7 \)
\( C = 7 – 0.5 = 6.5 = \frac{13}{2} \)
Step 3 — Write final expression:
Substitute \( C \) back into the general solution:
\( g(x) = \sin x + \frac{1}{2} e^{2x} + \frac{13}{2} \)
Final Answer: \( \boxed{g(x) = \sin x + \frac{1}{2} e^{2x} + \frac{13}{2}} \)