Integration

Indefinite integral

Indefinite integral of a function means finding the anti-derivative.

\({\displaystyle \int x^n\, dx = \frac{x^{n+1}}{n+1}+C}~~~(\mathrm{for}~n\ne-1)\)

\({\displaystyle \int x^{-1}\, dx =\ln |x| +C}\)

\({\displaystyle \int e^x\, dx=e^x +C}\)

Calculate the following integrals:

(a) \({\displaystyle \int (4x+5)\,dx}\)

(b) \({\displaystyle \int (x^2-6x+2)\,dx}\)

Solutions: (a) \(\cdots=2x^2+5x+C\); (b) \(\cdots=\frac{1}{3}x^3-3x^2+2x+C\); (c) \(\cdots=\frac{1}{4}x^4-\frac{2}{3}x^3+\frac{5}{2}x^2-7x+C\)

Calculate the following integrals (hint: expand the brackets first).

(a) \({\displaystyle \int (x+2)(x+4)\,dx}\)

(b) \({\displaystyle \int (x-1)(x+1)\,dx}\)

Solutions: (a) \(\cdots=\frac{1}{3}x^3+3x^2+8x+C\); (b) \(\cdots=\frac{1}{3}x^3-x+C\); (c) \(\cdots=3x^3+3x^2+x+C\)

Integrate:

(a) \({\displaystyle \int \left(6x^2+e^x\right)\,dx}\)

(b) \({\displaystyle \int \left(7e^x+\frac{1}{x}\right)\,dx}\)

(d) \({\displaystyle \int \left(x^5-\frac{6}{x^3}\right)\,dx}\)

Solutions: (a) \(\cdots=2x^3+e^x+C\); (b) \(\cdots=7e^x+\ln |x|+C\); (c) \(\cdots=\ln |x| -\frac{1}{x}+C\); (d) \(\cdots=\frac{1}{6}x^6+\frac{3}{x^2}+C\)

Integrate with respect to \(x\):

(a) \({\displaystyle \int (x+\sqrt{x}+1)\,dx}\)

(b) \({\displaystyle \int \frac{3}{\sqrt{x}}\,dx}\)

Solutions: (a) \(\cdots=\frac{1}{2}x^2+\frac{2}{3}\sqrt{x^3}+x+C\); (b) \(\cdots=6\sqrt{x}+C\); (c) \(\cdots=\frac{3}{4}\sqrt[3]{x^4}+\frac{3}{5}\sqrt[3]{x^5}+C\)

Find the equation of the function which has \(f'(x)=6x-4\) and \(f(1)=5\).
Solution: \(f(x)=3x^2-4x+6\)

Find the function \(f\) knowing that its derivative is \(f'(x)=2x+3\) and its graph passes through the point \(P(2,8)\).
Solution: \(f(x)=x^2+3x-2\)

A function has the value 8 at \(x=3\). Its derivative is \(f'(x)=x^2+4x-7\). Find the equation of this function.
Solution: \(f(x)=\frac{1}{3}x^3+2x^2-7x+2\)

A curve passes through the point \(P(1,4)\) and its gradient function is \(9x^2-5\). Write down the equation of this curve.
Solution: \(y=3x^3-5x+6\)

Find \(y\) and express it in terms of \(x\) (for \(x\gt0\)), given that \({\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x}}\) and \(y=3\) when \(x=1\).
Solution: \(y=\ln x+3\)

Definite integral

Definite integral is calculated as:

\({\displaystyle \int\limits_a^b f(x)\, dx = F(b)-F(a)}\)

Here \(F\) is the indefinite integral of \(f\):
\({\displaystyle F(x)=\int f(x)\, dx}\)

Evaluate each of the following integrals:

(a) \({\displaystyle \int\limits_{-1}^2 (-x^2+5)~ dx}\)

(b) \({\displaystyle \int\limits_0^2 (x^3-x+2)~dx }\)

Solutions: (a) \(\cdots=12\); (b) \(\cdots=6\); (c) \(\cdots=4\)

Evaluate each of the following integrals:

(a) \({\displaystyle \int\limits_1^2 x^{-2}~ dx}\)

(b) \({\displaystyle \int\limits_0^9 \sqrt{x}~ dx}\)

Solutions: (a) \(\cdots=\frac{1}{2}\); (b) \(\cdots=18\); (c) \(\cdots=e-1\approx1.72\)

Evaluate the integrals:

(a) \({\displaystyle \int\limits_{-2}^0 (x^3-4x)~ dx}\)

(b) \({\displaystyle \int\limits_0^2 (x^3-4x)~ dx}\)

Solutions: (a) \(\cdots=4\); (b) \(\cdots=-4\); (c) \(\cdots=0\)

Find the area of the figure enclosed by the graph of the function \(f(x)=x^2-x+1\), the \(x\)-axis and vertical lines \(x=-1\) and \(x=2\).
Solution: \(A=\frac{9}{2}=4.5\)

Find the area of the region between the graph of the function \(f(x)=e^x\) and the \(x\)-axis for \(0\leqslant x\leqslant 2\). Round the result to three significant figures.
Solution: \(A=6.39\)

Find the area of the region between the curve \(y=3\sqrt{x}\) and the \(x\)-axis on the interval \([1,4]\).
Solution: \(A=14\)

Find the area of the region between the curve \(y=9-x^2\) and the \(x\)-axis.
Solution: \(A=36\)

Find the area of the region enclosed by the graph of the function \(f(x)=x^3+2x^2\) and the \(x\)-axis.
Solution: \(A=\frac{4}{3}\)

Find the area of the region enclosed by the curve \(y=x^2\) and the straight line \(y=x+2\).
Solution: \(A=\frac{9}{2}=4.5\)

Find the area between the curves \(y=x^2-4x-2\) and \(y=4-x^2\).
Solution: \(A=21\frac{1}{3}\)

Find the area of the figure between the graphs of the functions \(f(x)=x^2-1\) and \(g(x)=5-x^2\). Give the result in the exact form.
Solution: \(A=8\sqrt{3}\)

Find the area of the region enclosed by the graphs of the functions \(f(x)=\sqrt{3x}\) and \(g(x)=x^2-2x\).
Solution: \(A=6\)

Find the area of the region enclosed by the graphs of the functions \(f(x)=\sqrt[\scriptstyle 3]{x}\) and \(g(x)=x^4\).
Solution: \(A=0.55\)

Find the area of the region enclosed by the curve \(\displaystyle f(x)=\frac{1}{\,x\,}\) and the straight line \(\displaystyle y=-x+\frac{5}{2}\).
Solution: \(A\approx0.489\)

Find the area of the region enclosed by the curve \(\displaystyle f(x)=\frac{1}{x^2+1}\) and the straight line \(\displaystyle y=\frac{1}{10}\).
Solution: \(A\approx1.90\)

Find the area of the region enclosed by the curves \(\displaystyle f(x)=\frac{4}{\,x\,}\) and \(g(x)=(x-3)^2\).
Solution: \(A\approx2.55\)

Consider the functions \(f(x)=x^4-3x\) and \(g(x)=3-x^2\).

(a) Write down the intersection points of \(f\) and \(g\).

(b) Find the area of the region bounded by the graphs of \(f\) and \(g\).

Solutions: (a) \(P_1(-0.729,2.47),~ P_2(1.51,0.709)\); (b) \(A\approx6.45\)

Let \(f(x)=x^3+3x^2+x+3\).

(a) Write down the equation of the tangent to \(f\) at \(x=0\).

(b) Find the area of the region bounded by the graph of \(f\) and this tangent.

Solutions: (a) \(y=x+3\); (b) \(A=6.75\)

Let \(f(x)=\frac{1}{2}x^3-2x^2+2x\).

(a) Write down the equation of the tangent to \(f\) at \(x=0\).

(b) Find the area of the region bounded by the graph of \(f\) and this tangent.

Solutions: (a) \(y=2x\); (b) \(A\approx10.7\)

Let \(f(x)=\frac{1}{3}x^4-x^3\).

(a) Write down the equation of the normal to \(f\) at point \(A(3,0)\).

This normal intersects the graph of the function at \(A\) and at another point \(B\).

(b) Write down the coordinates of \(B\).

Solutions: (a) \(y=-\frac{1}{9}x+\frac{1}{3}\); (b) \(B(-0.693,0.410)\); (c) Area \(\approx4.74\)

Find the area of the region enclosed by the curve \(y=\sqrt{x+4}\), the straight line \(y=2-x\) and the \(x\)-axis.
Solution: \(A\approx7.33\)

Consider the function \(f(x)=x^2-4x+6\).

(a) Write down the equation of the tangent to \(f\) at \(x=0\).

(b) Write down the equation of the tangent to \(f\) at \(x=3\).

(d) Find the area of the region bounded by the graph of \(f\) and both tangents.

Solutions: (a) \(y=-4x+6\); (b) \(y=2x-3\); (c) \(P(1.5,0)\); (d) Area \(=\frac{9}{4}=2.25\)

Trapezoidal formula

Trapezoidal formula can be used to approximate the definite integral (or the area):

\(A\approx \frac{1}{2}h(y_0+2y_1+2y_2+2y_3+\cdots+y_n)\)

Here \(h=\frac{b-a}{n}\)

Let \(f(x)=\frac{1}{4}x^2+1\). Use the trapezoidal formula to find the approximate area of the region between the graph and the horizontal axis on the interval \([0,4]\). Divide this interval in four parts.
Solution: \(A\approx9.5\)

Let \(A={\displaystyle \int\limits_1^9 \sqrt{x}~ dx}\). Find the approximate value of this integral using the trapezoidal formula.

(a) Divide the interval in four parts.

(b) Divide the interval in eight parts.

Solutions: (a) \(A\approx17.23\); (b) \(A\approx17.31\); (c) \(A=17\frac{1}{3}\)

A table of values of the function \(y=f(x)\) is given:

\(x\)	0	0.5	1	1.5	2	2.5	3
\(y\)	1.75	2.25	3.25	2.75	2.5	2	1.25

Use the trapezoidal formula to find the area of the region under the graph of the function between \(x=1\) and \(x=3\).
Solution: \(A\approx4.75\)