Mixed practice: integrals and areas Problem set A

Find the exact value of \(\int_0^{\ln 5} 4e^{-2x}\,dx\).

Find the exact value of \(\int_{-12}^{-3}\frac{8}{x}\,dx\), giving your answer as a multiple of a single logarithm.

Find the exact value of each integral.

1. (a) \(\int_1^6\frac6{3x+2}\,dx\);
2. (b) \(\int_1^4\frac1{x-7}\,dx\).

Show that the value of \(\int_k^{3k}\frac1x\,dx\), where \(k>0\), is independent of k.

In each part, a is positive. Find a.

1. (a) \(\int_0^a(x^2+2x)\,dx=18\);
2. (b) \(\int_a^{2a}(3x+2)\,dx=46.5\).

Use the properties of integrals.

1. (a) Given that \(\int_1^5f(x)\,dx=9\), find \(\int_1^5(4f(x)-3)\,dx\);
2. (b) Given that \(\int_0^8f(x)\,dx=18\), find \(\int_0^43f(2x)\,dx\).

Use reverse differentiation.

1. (a) Differentiate \(x\ln(2x)\) with respect to x;
2. (b) hence find the exact value of \(\int_1^3\ln(2x)\,dx\).

Find the exact value of \(\int_0^2x\sqrt{x^2+5}\,dx\).

The curves \(y=e^{2x}\) and \(y=e^{-x}\) meet at \((0,1)\). Find the exact area of the shaded region, which lies below both curves, above the x-axis, from \(x=-1\) to \(x=1\).

A flower bed is the region enclosed by the curves \(y=e^{-x^2}\) and \(y=e^{-2x}\), where x and y are in metres. The curves meet at A on the y-axis and at B.

1. (a) Find the coordinates of A and B.
2. (b) Find the area of the flower bed.

The diagram shows the graphs of \(y=\sin x\) and \(y=\cos x\) for \(0\le x\le\frac\pi2\).

1. (a) Write down the exact coordinates of the point where the graphs meet.
2. (b) Find the exact area of the shaded region, between the two graphs and the lines \(x=0\) and \(x=\frac\pi2\).

Let \(f(x)=\sin(x^2)\) and \(g(x)=\cos(x^2)\), for \(0\le x\le1.5\).

1. (a) Sketch the graphs of f and g on the same axes.
2. (b) Find the area enclosed by the two graphs and the y-axis.

Show that the area enclosed by the graph of \(y=\sin3x\) and the x-axis for \(0\le x\le\frac{2\pi}3\) is \(\frac43\).

The diagram shows part of the graph of \(y=f(x)\), which crosses the x-axis at a, b, c and d. The diagram is not drawn to scale. Given that \(\int_a^b f(x)\,dx=7\), \(\int_a^d f(x)\,dx=5\) and \(\int_a^d|f(x)|\,dx=21\), find

1. (a) \(\int_b^d f(x)\,dx\);
2. (b) \(\int_a^c f(x)\,dx\);
3. (c) the area of the region between the curve and the x-axis from b to c.