Question

6. Graph the integrand and use areas to evaluate the integral. Find the domain of the integrand ſ(3+v1- x?)dx

Answer 1

|(71–72 +3) dz Apply linearity: = (v1 – cºde + 3/1dx Now solving: 1 – x2 da Perform trigonometric substitution: Substitute x = sin(u) zu = arcsin(x), dx = cos(u) du (steps): = | cos(u)v1 – sin’(u) du Simplify using 1 – sin(u) sin?(u) = cos? (u): = | cos”(u) de cos-1(u) si | cosulu) du = = = */ cos+ ?(u) du n with n= 2: cos(u) sin(u) + 2 1 du 1 du Apply constant rule: =U 1 2 u Plug in solved integrals: cos(u) sin(u) + 1 du 2 cos(u) sin(u) + 2 2 Undo substitution u = arcsin(x), use: sin(arcsin(x)) = x cos(arcsin(x)) = V1 – x2 arcsin(x) XV1 - x2 + 2 2 Now solving: 1 da Use previous result: = X Plug in solved integrals: JV1-0'da +3/1de 1 arcsin(x) XV1 – x2 + + 3x 2 2 1 S f(x) dx -1 + 12 2 Approximation: 7.570796326794897 Scanned with CamScanner

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