Exercises - Fundamental Theorem of Calculus

  1. Integrate

    1. $\displaystyle{\int_0^1 (x^3+x^2) \ dx}$

    2. $\displaystyle{\int_0^{\pi/2} \cos x \ dx}$

    3. $\displaystyle{\int_1^2 (4x^2-x) \ dx}$

    4. $\displaystyle{\int_0^{\pi/4} \frac{1}{3} \sec^2 x \ dx}$

    5. $\displaystyle{\int_2^5 \frac{x+2}{\sqrt{x-1}} \ dx}$

    6. $\displaystyle{\int_1^9 \frac{1}{\sqrt{x}(1+\sqrt{x})^2} \ dx}$

    7. $\displaystyle{\int_0^{\pi/4} \tan^3 x \cos x \ dx}$

    8. $\displaystyle{\int_0^{\pi/3} \sec x \tan x \ dx}$

    9. $\displaystyle{\int_0^{\sqrt{\pi}} x \sin(x^2) \ dx}$

    10. $\displaystyle{\int_0^{\pi/3} \sin(2x) \cos^2(2x) \ dx}$

    11. $\displaystyle{\int_{\pi/6}^{\pi/2} \frac{1-\sin^2 x}{\cos x} \ dx}$

    12. $\displaystyle{\int_0^{4/3} \sqrt{1 + \frac{9}{4}x} \ dx}$

    13. $\displaystyle{\int_0^1 x^2 \sqrt{4+5x^3} \ dx}$

    1. $\displaystyle{\int_0^2 \frac{1}{(3x+2)^2} \ dx}$

    2. $\displaystyle{\int_0^2 \frac{4y}{\sqrt{25-4y^2}} \ dy}$

    3. $\displaystyle{\int_0^1 18x \sqrt{3x^2+1} \ dx}$

    4. $\displaystyle{\int_0^{\sqrt{2}/4} \frac{t}{\sqrt{1-4t^2}} \ dt}$

    5. $\displaystyle{\int_4^8 \frac{3t}{\sqrt{t^2-15}} \ dt}$

    6. $\displaystyle{\int_0^{\pi/2} \sec^2 \left( \frac{x}{2} \right) \ dx}$

    7. $\displaystyle{\int_0^{\pi/4} \cos^2(2x)\sin(2x) \ dx}$

    8. $\displaystyle{\int_0^{1/2} \frac{\textrm{Arctan } (2x)}{1+4x^2} \ dx}$

    9. $\displaystyle{\int_0^{1/\sqrt{2}} \frac{x}{\sqrt{1-4x^4}} \ dx}$

    10. $\displaystyle{\int_1^e \frac{\ln \sqrt{x}}{x} \ dx}$

    11. $\displaystyle{\int_3^6 \frac{y}{3\sqrt{y^2-8}} \ dy }$


  2. Find the area between the graph of the given function and the $x$-axis over the given interval

    1. $y=x^3 \quad ; \quad [-1,3]$

    2. $y=\sin x \quad ; \quad [-\frac{\pi}{2},\frac{\pi}{2}]$

    3. $y=-x^2+4 \quad ; \quad [-4,2]$

    4. $y=x^3-3x^2 \quad ; \quad [1,4]$

    5. $y=\frac{1}{4}x^4-x^2 \quad ; \quad [-1,1]$

    6. $y=3x^4+4x^3 \quad ; \quad [-1,1]$