This problem set covers every topic from the Integration Basics lesson — indefinite integrals, definite integrals, the Fundamental Theorem of Calculus, u-substitution, and area between curves. Work through each problem on your own, then click Show Solution to check your work step by step.
📚 Table of Contents
- Indefinite Integrals & Basic Rules
- Definite Integrals & FTC
- U-Substitution
- Trig Integrals
- Exponential & Log Integrals
- Area Between Curves
- Mixed & Challenge Problems
1. Indefinite Integrals & Basic Rules
Problem 1. \(\displaystyle\int (6x^5 – 4x^3 + 2x – 1)\,dx\)
👁️ Show Solution
$$x^6 – x^4 + x^2 – x + C$$Problem 2. \(\displaystyle\int \frac{1}{x^5}\,dx\)
👁️ Show Solution
Rewrite as \(x^{-5}\):
$$\frac{x^{-4}}{-4} + C = \frac{-1}{4x^4} + C$$Problem 3. \(\displaystyle\int \sqrt[4]{x^3}\,dx\)
👁️ Show Solution
Rewrite as \(x^{3/4}\):
$$\frac{x^{7/4}}{7/4} + C = \frac{4}{7}x^{7/4} + C$$Problem 4. \(\displaystyle\int \left(4\sqrt{x} – \frac{3}{x^2} + 5\right)\,dx\)
👁️ Show Solution
Rewrite as \(4x^{1/2} – 3x^{-2} + 5\):
$$\frac{8}{3}x^{3/2} + \frac{3}{x} + 5x + C$$Problem 5. \(\displaystyle\int (x+1)(x-3)\,dx\)
👁️ Show Solution
Expand first: \((x+1)(x-3) = x^2 – 2x – 3\)
$$\frac{x^3}{3} – x^2 – 3x + C$$Problem 6. \(\displaystyle\int \frac{x^3 + 2x}{x}\,dx\)
👁️ Show Solution
Divide first: \(\frac{x^3 + 2x}{x} = x^2 + 2\)
$$\frac{x^3}{3} + 2x + C$$2. Definite Integrals & FTC
Problem 7. Evaluate \(\displaystyle\int_1^3 (2x^2 – 3)\,dx\)
👁️ Show Solution
$$F(x) = \frac{2x^3}{3} – 3x$$ $$F(3) – F(1) = (18 – 9) – \left(\frac{2}{3} – 3\right) = 9 + \frac{7}{3} = \frac{34}{3}$$Problem 8. Evaluate \(\displaystyle\int_0^{\pi} \sin(x)\,dx\)
👁️ Show Solution
$$F(x) = -\cos(x)$$ $$F(\pi) – F(0) = -\cos(\pi) – (-\cos(0)) = 1 + 1 = 2$$Problem 9. Evaluate \(\displaystyle\int_1^e \frac{1}{x}\,dx\)
👁️ Show Solution
$$F(x) = \ln|x|$$ $$F(e) – F(1) = \ln(e) – \ln(1) = 1 – 0 = 1$$Problem 10. Find \(\displaystyle\frac{d}{dx}\!\left[\int_3^x (t^2 + e^t)\,dt\right]\)
👁️ Show Solution
By FTC Part 1:
$$x^2 + e^x$$Problem 11. Find \(\displaystyle\frac{d}{dx}\!\left[\int_0^{x^2} \cos(t)\,dt\right]\)
👁️ Show Solution
Upper limit is \(x^2\), apply Chain Rule:
$$\cos(x^2) \cdot 2x = 2x\cos(x^2)$$Problem 12. Evaluate \(\displaystyle\int_0^2 |x – 1|\,dx\)
👁️ Show Solution
Split at \(x = 1\) where the expression changes sign:
$$\int_0^1 (1-x)\,dx + \int_1^2 (x-1)\,dx$$ $$= \left[x – \frac{x^2}{2}\right]_0^1 + \left[\frac{x^2}{2} – x\right]_1^2 = \frac{1}{2} + \frac{1}{2} = 1$$3. U-Substitution
Problem 13. \(\displaystyle\int 4x(x^2 + 3)^6\,dx\)
👁️ Show Solution
Let \(u = x^2 + 3\), \(du = 2x\,dx\), so \(4x\,dx = 2\,du\)
$$\int 2u^6\,du = \frac{2u^7}{7} + C = \frac{2(x^2+3)^7}{7} + C$$Problem 14. \(\displaystyle\int \frac{x}{\sqrt{x^2 + 1}}\,dx\)
👁️ Show Solution
Let \(u = x^2 + 1\), \(du = 2x\,dx\), so \(x\,dx = \frac{du}{2}\)
$$\int \frac{1}{\sqrt{u}} \cdot \frac{du}{2} = \frac{1}{2} \cdot 2\sqrt{u} + C = \sqrt{x^2+1} + C$$Problem 15. \(\displaystyle\int \frac{\ln(x)}{x}\,dx\)
👁️ Show Solution
Let \(u = \ln(x)\), \(du = \frac{1}{x}\,dx\)
$$\int u\,du = \frac{u^2}{2} + C = \frac{[\ln(x)]^2}{2} + C$$Problem 16. \(\displaystyle\int \tan(x)\,dx\)
👁️ Show Solution
Rewrite as \(\displaystyle\int \frac{\sin(x)}{\cos(x)}\,dx\). Let \(u = \cos(x)\), \(du = -\sin(x)\,dx\)
$$\int \frac{-du}{u} = -\ln|u| + C = -\ln|\cos(x)| + C$$Problem 17. Evaluate \(\displaystyle\int_0^1 x\,e^{x^2}\,dx\)
👁️ Show Solution
Let \(u = x^2\), \(du = 2x\,dx\). Change limits: \(x=0 \Rightarrow u=0\), \(x=1 \Rightarrow u=1\)
$$\int_0^1 \frac{e^u}{2}\,du = \frac{1}{2}\left[e^u\right]_0^1 = \frac{1}{2}(e – 1)$$Problem 18. \(\displaystyle\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx\)
👁️ Show Solution
Let \(u = \sqrt{x}\), \(du = \frac{1}{2\sqrt{x}}\,dx\), so \(\frac{dx}{\sqrt{x}} = 2\,du\)
$$\int e^u \cdot 2\,du = 2e^u + C = 2e^{\sqrt{x}} + C$$4. Trig Integrals
Problem 19. \(\displaystyle\int (4\cos(x) – 3\sin(x))\,dx\)
👁️ Show Solution
$$4\sin(x) + 3\cos(x) + C$$Problem 20. \(\displaystyle\int \sin(4x)\,dx\)
👁️ Show Solution
Let \(u = 4x\), \(du = 4\,dx\):
$$-\frac{\cos(4x)}{4} + C$$Problem 21. \(\displaystyle\int \sec^2(3x)\,dx\)
👁️ Show Solution
Let \(u = 3x\), \(du = 3\,dx\):
$$\frac{\tan(3x)}{3} + C$$Problem 22. Evaluate \(\displaystyle\int_0^{\pi/2} \cos(x)\,dx\)
👁️ Show Solution
$$[\sin(x)]_0^{\pi/2} = \sin\!\left(\frac{\pi}{2}\right) – \sin(0) = 1 – 0 = 1$$Problem 23. \(\displaystyle\int \sin^3(x)\cos(x)\,dx\)
👁️ Show Solution
Let \(u = \sin(x)\), \(du = \cos(x)\,dx\)
$$\int u^3\,du = \frac{u^4}{4} + C = \frac{\sin^4(x)}{4} + C$$5. Exponential & Log Integrals
Problem 24. \(\displaystyle\int 5e^{3x}\,dx\)
👁️ Show Solution
$$\frac{5e^{3x}}{3} + C$$Problem 25. \(\displaystyle\int \frac{3}{2x+1}\,dx\)
👁️ Show Solution
Let \(u = 2x+1\), \(du = 2\,dx\):
$$\frac{3}{2}\ln|2x+1| + C$$Problem 26. \(\displaystyle\int 4^x\,dx\)
👁️ Show Solution
$$\frac{4^x}{\ln(4)} + C$$Problem 27. \(\displaystyle\int \frac{x^2 + 1}{x}\,dx\)
👁️ Show Solution
Divide first: \(\frac{x^2+1}{x} = x + \frac{1}{x}\)
$$\frac{x^2}{2} + \ln|x| + C$$Problem 28. Evaluate \(\displaystyle\int_0^1 e^{2x}\,dx\)
👁️ Show Solution
$$\left[\frac{e^{2x}}{2}\right]_0^1 = \frac{e^2}{2} – \frac{1}{2} = \frac{e^2 – 1}{2}$$6. Area Between Curves
Problem 29. Find the area between \(f(x) = x + 2\) and \(g(x) = x^2\) on \([-1, 2]\)
👁️ Show Solution
On \([-1, 2]\): \(x + 2 \geq x^2\)
$$A = \int_{-1}^2 (x + 2 – x^2)\,dx = \left[\frac{x^2}{2} + 2x – \frac{x^3}{3}\right]_{-1}^2$$ $$= \left(2 + 4 – \frac{8}{3}\right) – \left(\frac{1}{2} – 2 + \frac{1}{3}\right) = \frac{10}{3} + \frac{7}{6} = \frac{9}{2}$$Problem 30. Find the area enclosed by \(f(x) = x^2 – 2x\) and the x-axis
👁️ Show Solution
Find intersections: \(x^2 – 2x = 0 \Rightarrow x(x-2) = 0 \Rightarrow x = 0,\ x = 2\)
On \([0, 2]\): \(f(x) \leq 0\), so integrate \(|f(x)|\)
$$A = \int_0^2 (2x – x^2)\,dx = \left[x^2 – \frac{x^3}{3}\right]_0^2 = 4 – \frac{8}{3} = \frac{4}{3}$$Problem 31. Find the area enclosed by \(f(x) = \sin(x)\) and \(g(x) = \cos(x)\) on \(\left[\frac{\pi}{4},\ \frac{5\pi}{4}\right]\)
👁️ Show Solution
On \(\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]\): \(\sin(x) \geq \cos(x)\)
$$A = \int_{\pi/4}^{5\pi/4} (\sin x – \cos x)\,dx = \left[-\cos x – \sin x\right]_{\pi/4}^{5\pi/4}$$ $$= \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\right) – \left(-\frac{\sqrt{2}}{2} – \frac{\sqrt{2}}{2}\right) = \sqrt{2} + \sqrt{2} = 2\sqrt{2}$$7. Mixed & Challenge Problems 🔥
Problem 32. \(\displaystyle\int \frac{\cos(\ln x)}{x}\,dx\)
👁️ Show Solution
Let \(u = \ln(x)\), \(du = \frac{1}{x}\,dx\)
$$\int \cos(u)\,du = \sin(u) + C = \sin(\ln x) + C$$Problem 33. Evaluate \(\displaystyle\int_0^4 \frac{x}{\sqrt{x^2+9}}\,dx\)
👁️ Show Solution
Let \(u = x^2 + 9\), \(du = 2x\,dx\). Change limits: \(x=0 \Rightarrow u=9\), \(x=4 \Rightarrow u=25\)
$$\int_9^{25} \frac{du}{2\sqrt{u}} = \left[\sqrt{u}\right]_9^{25} = 5 – 3 = 2$$Problem 34. \(\displaystyle\int \frac{(\ln x)^3}{x}\,dx\)
👁️ Show Solution
Let \(u = \ln(x)\), \(du = \frac{1}{x}\,dx\)
$$\int u^3\,du = \frac{u^4}{4} + C = \frac{(\ln x)^4}{4} + C$$Problem 35. Find the area enclosed by \(f(x) = x^3\) and \(g(x) = x\)
👁️ Show Solution
Intersections: \(x^3 = x \Rightarrow x(x^2-1) = 0 \Rightarrow x = -1, 0, 1\)
By symmetry, compute on \([0,1]\) and double. On \([0,1]\): \(x \geq x^3\)
$$A = 2\int_0^1 (x – x^3)\,dx = 2\left[\frac{x^2}{2} – \frac{x^4}{4}\right]_0^1 = 2 \cdot \frac{1}{4} = \frac{1}{2}$$Problem 36. \(\displaystyle\int \frac{2x^3 + 3x}{x^4 + 3x^2 + 1}\,dx\)
👁️ Show Solution
The numerator is half the derivative of the denominator:
Let \(u = x^4 + 3x^2 + 1\), \(du = (4x^3 + 6x)\,dx = 2(2x^3 + 3x)\,dx\)
So \((2x^3 + 3x)\,dx = \frac{du}{2}\)
$$\int \frac{du/2}{u} = \frac{1}{2}\ln|u| + C = \frac{1}{2}\ln|x^4 + 3x^2 + 1| + C$$🔗 Continue Your Calculus 1 Journey
👉 Back to Integration Basics Lesson — Coming soon!
👉 Derivatives Applications Lesson — Coming soon!
👉 Derivatives — Rules & Techniques — Coming soon!
👉 Limits & Continuity Lesson — Coming soon!
👉 Final Exam Extra Practice Problem Set — Coming soon!
👉 Full Calculus 1 Study Guide — Coming soon!
