The Superhero Integral Challenge
Storyboard Text
- Got it! Here's the solution to your integral problem.
- Step 1: Use the power rule to find the antiderivative of 3x^2.The power rule states that the antiderivative of x^n is (x^(n+1))/(n+1), so:∫ 3x^2 dx = x^3 + C1where C1 is the constant of integration.
- Step 2: Use the exponential rule to find the antiderivative of e^x.The exponential rule states that the antiderivative of e^x is e^x, so:\∫ e^x dx = e^x + C2where C2 is the constant of integration.
- Step 3: Use the trigonometric rule to find the antiderivative of 5sin(x).The trigonometric rule states that the antiderivative of sin(x) is -cos(x), so:∫ 5sin(x) dx = -5cos(x) + C3where C3 is the constant of integration.
- Step 4: Use substitution and the power rule to find the antiderivative of√(2x+1).Let u = 2x+1, then du/dx = 2 and dx = du/2.Substituting u and dx in the integral:∫ √(2x+1) dx = ∫ √u (du/2).Using the power rule for the integral of √u:∫ √u (du/2) = (2/3)u^(3/2) + C4.Substituting back u = 2x+1, we get:∫ √(2x+1) dx = (2/3)(2x+1)^(3/2) + C4where C4 is the constant of integration.
- Step 5: Combine the antiderivativesPutting all the antiderivatives together, we get:∫ √(2x+1) + e^x + 5sin(x) + 3x^2 dx = (2/3)(2x+1)^(3/2) + e^x - 5cos(x) + x^3 + Cwhere C is the constant of integration that combines all the previous constants.
- I have no doubt you will, Alex. Keep up the good work.
- No problem, Alex. As a superhero, it's my duty to use my powers to help others. Just keep using your knowledge of math to keep the city safe.
- I will, Integral Avenger. And who knows, maybe one day I'll be able to solve these problems on my own.
- That's incredible! Thank you so much for your help, Dark Integral Avenger.
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