= x e x – ∫ dx = xe x – e x = e x (x – 1) Let, The first function = f(x) = x and the second function = g(x) = e x. To solve I 1, we will use the rule of integration by parts. Where, I 1 = ∫ x e x dx and I 2 = ∫ sin (πx/4) dx Solving I 1 Question 3: Evaluate ∫ 0 1 (xe x + sin (πx/4) dxĪnswer : Let I = ∫ 0 1 (xe x + sin (πx/4) dx. Using the standard integral ∫ dx / (√) = sin –1 (x/a), we get Question 1: Evaluate ∫ 0 1 dx / (√)Īnswer : Let I = ∫ 0 1 dx / (√). To solve the indefinite integral, let’s substitute sin2t = u, so that 2 cos2t dt = du or cos2t dt = ½ du. Solution: Let I = ∫ 0 π/4 sin 32t cos2t dt Using the rules of partial fractions, we get Solution: Let I = ∫ 1 2 x dx / (x + 1)(x + 2) Hence, using the second fundamental theorem of calculus, we get Let’s substitute (30 – x 3/2) = t, so that – 3/2 √x dx = dt or √x dx = – 2/3 dt. Solution: Let I = ∫ 4 9 dxįirst, we find the anti-derivative of the integrand. Using the second fundamental theorem of calculus, we get Now, the indefinite integral, ∫ x 2 dx = x 3/3 = F(x) That’s it! Let’s look at some examples now. There is no need to keep the integration constant C because it disappears while evaluating the value of the definite integral. Find the indefinite integral ∫ f(x) dx.Two simple steps can help you calculate ∫ a b f(x) dx as shown below: FUNDAMENTAL THEOREM CALCULUS CALCULATOR DOWNLOADYou can download Integrals Cheat Sheet by clicking on the download button below
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