Exam 13: Integral Calculus

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Find the integral. - x45x5+3dx\int \frac{x^{4}}{5 x^{5}+3} d x

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Evaluate the definite integral. - 04et(10+et)3dt\int_{0}^{4} \frac{\mathrm{e}^{\mathrm{t}}}{\left(10+\mathrm{e}^{\mathrm{t}}\right)^{3}} d t

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Provide an appropriate response. -If we use u=x2u=x^{2} as a substitution to find xex2dxx e^{x^{2}} d x , then which of the following would be the correct results? i. (eu2)du\int\left(\frac{e^{u}}{2}\right) d u \quad ii) (2eu)du\int\left(2 e^{u}\right) d u \quad iii) (ueu)du\int\left(u e^{u}\right) d u

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Find the particular solution of the differential equation. - dydx=4x;y=13\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{\mathrm{x}} ; \mathrm{y}=13 when x=1\mathrm{x}=1

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Find the area of the shaded region. -Find the area of the shaded region. -

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Find the integral. - 3(2t+5)3dt\int 3(2 t+5)^{3} d t

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Find the integral. - x5+1xdx\int \frac{x^{5}+1}{x} d x

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Find the area bounded by the given curves. - y=5xy=\frac{5}{x} and y=1+5xx2;[1,5]y=1+5 x-x^{2} ;[1,5]

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Use integration by parts to find the integral. - 3xexdx\int 3 x e^{x} d x

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Find the integral. - 7pe(5p2)dp\int 7 p e^{\left(5 p^{2}\right)} d p

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Find the area bounded by the given curves. - y=2xx2,y=2x4y=2 x-x^{2}, y=2 x-4

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Find the given indefinite integral. - ln(7x+8)dx\int \ln (7 x+8) d x

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Work the exercise. -Find the total revenue function R(x)R(x) (in thousands of dollars) if the marginal revenue (in thousands of dollars per unit) at a production level of xx units is R=49x+3ex(0x7)R^{\prime}=\frac{49 x+3}{e^{x}}(0 \leq x \leq 7) , and R(0)=0R(0)=0 .

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Use integration by parts to find the integral. - ln6xx3dx\int \frac{\ln 6 x}{x^{3}} d x

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Evaluate the integral. - 14x1/2dx\int_{1}^{4} x^{1 / 2} d x

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Find the particular solution of the differential equation. - 2dydx4xy=8x;y=82 \frac{d y}{d x}-4 x y=8 x ; y=8 when x=0x=0

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Solve the problem. -The work WW (in joules) done by a force FF (in newtons) moving an object through a distance xx (in meters) is given by W=Fdx\mathrm{W}=\int \mathrm{Fdx} . Find a formula for W\mathrm{W} , if F=kx\mathrm{F}=\mathrm{kx} and k\mathrm{k} is a constant.

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Find the integral. - (x6+6x)dx\int\left(\frac{x}{6}+\frac{6}{x}\right) d x

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Find the integral. - dx(5+lnx)x\int \frac{d x}{(5+\ln x) x}

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Find the integral. - 6x7dx\int \sqrt{6 x-7} d x

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