int[e, e^4] [1/(x ln(x)] dx, deriv. of int[e^x, 0] [sin^2(t)] dt, (d^2 f)/dx = ...

[eatfish521]

\(\displaystyle \mbox{1. Compute the integral: }\, \)\(\displaystyle \displaystyle \int_e^{e^4}\, \)\(\displaystyle \dfrac{1}{x\, \cdot\, \ln(x)}\, dx\)

\(\displaystyle \mbox{2. Find the derivative of the function: }\, f(x)\,=\, \)\(\displaystyle \displaystyle \int_{e^x}^0\, \left(\sin(t)\right)^2\, dt\)

\(\displaystyle \mbox{3. Estimate the area under the graph of }\, f(x)\, =\, 1\, +\, x^2\, \mbox{ from }\, x\, =\, -1\, \mbox{ to }\, x\, =\, 3,\,\)\(\displaystyle \mbox{ using four rectangles and right endpoints.}\)

\(\displaystyle \mbox{4. Find }\, f(x)\, \mbox{ if }\, \dfrac{d^2 f}{dx^2}\, =\, 6\, -\, 18\, \cdot\, x,\, f(0)\, =\, 6,\, \mbox{ and }\, f(2)\, =\, 12.\)

\(\displaystyle \mbox{5. Find the total area between the curve }\, y\, =\, x^2\, -\, 2\, \cdot\, x\, \mbox{ and the }\, x-\mbox{axis on the interval }\, \left[1,\, 3\right].\)

\(\displaystyle \mbox{6. Find }\, \)\(\displaystyle \displaystyle \int_{-2}^2\, \left(2\, \cdot\, f(x)\, +\, 3\, \cdot\, g(x)\right)\, dx \mbox{ if }\, f(x)\, \mbox{ is an even function such that }\) \(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, 3,\, \mbox{ and }\, g(x)\, \mbox{ is such that }\, \)\(\displaystyle \displaystyle \int_{-2}^4\, g(x)\, dx\, =\, -3\, \mbox{ and }\,\)\(\displaystyle \displaystyle \int_2^4\, g(x)\, dx\, =\, -6\)

Could somebody help me to solve those questions？ Thanks so much！

 

[Deleted member 4993]

\(\displaystyle \mbox{1. Compute the integral: }\, \)\(\displaystyle \displaystyle \int_e^{e^4}\, \)\(\displaystyle \dfrac{1}{x\, \cdot\, \ln(x)}\, dx\)

\(\displaystyle \mbox{2. Find the derivative of the function: }\, f(x)\,=\, \)\(\displaystyle \displaystyle \int_{e^x}^0\, \left(\sin(t)\right)^2\, dt\)

\(\displaystyle \mbox{3. Estimate the area under the graph of }\, f(x)\, =\, 1\, +\, x^2\, \mbox{ from }\, x\, =\, -1\, \mbox{ to }\, x\, =\, 3,\) \(\displaystyle \mbox{ using four rectangles and right endpoints.}\)

\(\displaystyle \mbox{4. Find }\, f(x)\, \mbox{ if }\, \dfrac{d^2 f}{dx^2}\, =\, 6\, -\, 18\, \cdot\, x,\, f(0)\, =\, 6,\, \mbox{ and }\, f(2)\, =\, 12.\)

\(\displaystyle \mbox{5. Find the total area between the curve }\, y\, =\, x^2\, -\, 2\, \cdot\, x\, \mbox{ and the }\, x-\mbox{axis on the interval }\, \left[1,\, 3\right].\)

\(\displaystyle \mbox{6. Find }\, \)\(\displaystyle \displaystyle \int_{-2}^2\, \left(2\, \cdot\, f(x)\, +\, 3\, \cdot\, g(x)\right)\, dx \mbox{ if }\, f(x)\, \mbox{ is an even function such that }\) \(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, 3,\, \mbox{ and }\, g(x)\, \mbox{ is such that }\) \(\displaystyle \displaystyle \int_{-2}^4\, g(x)\, dx\, =\, -3\, \mbox{ and }\) \(\displaystyle \displaystyle \int_2^4\, g(x)\, dx\, =\, -6\)

Could somebody help me to solve those questions？ Thanks so much！

Yes .... we can but...

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[Prove It]

\(\displaystyle \mbox{1. Compute the integral: }\, \)\(\displaystyle \displaystyle \int_e^{e^4}\, \)\(\displaystyle \dfrac{1}{x\, \cdot\, \ln(x)}\, dx\)

\(\displaystyle \mbox{2. Find the derivative of the function: }\, f(x)\,=\, \)\(\displaystyle \displaystyle \int_{e^x}^0\, \left(\sin(t)\right)^2\, dt\)

\(\displaystyle \mbox{3. Estimate the area under the graph of }\, f(x)\, =\, 1\, +\, x^2\, \mbox{ from }\, x\, =\, -1\, \mbox{ to }\, x\, =\, 3,\) \(\displaystyle \mbox{ using four rectangles and right endpoints.}\)

\(\displaystyle \mbox{4. Find }\, f(x)\, \mbox{ if }\, \dfrac{d^2 f}{dx^2}\, =\, 6\, -\, 18\, \cdot\, x,\, f(0)\, =\, 6,\, \mbox{ and }\, f(2)\, =\, 12.\)

\(\displaystyle \mbox{5. Find the total area between the curve }\, y\, =\, x^2\, -\, 2\, \cdot\, x\, \mbox{ and the }\, x-\mbox{axis on the interval }\, \left[1,\, 3\right].\)

\(\displaystyle \mbox{6. Find }\, \)\(\displaystyle \displaystyle \int_{-2}^2\, \left(2\, \cdot\, f(x)\, +\, 3\, \cdot\, g(x)\right)\, dx \mbox{ if }\, f(x)\, \mbox{ is an even function such that }\)

. . . . .\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, 3,\, \mbox{ and }\, g(x)\, \mbox{ is such that }\)\(\displaystyle \displaystyle \int_{-2}^4\, g(x)\, dx\, =\, -3\, \mbox{ and }\) \(\displaystyle \displaystyle \int_2^4\, g(x)\, dx\, =\, -6\)

Could somebody help me to solve those questions？ Thanks so much！

Some general hints:

1. Use the substitution u = ln(x) => du = (1/x) dx.

2. Use the second fundamental theorem of calculus.

3. Draw a graph, draw in the rectangles, evaluate their areas, add them up.

4. You have the second derivative, so to get the original function you need to integrate twice. You will have two integration constants, which is why you are given two points.

5. Do you know how to do a definite integral?