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About Integral Calculus AI

An AI solver for integral calculus can help solve various types of integration problems, including standard integration, definite integrals, and integration using different substitutions such as algebraic, trigonometric, and hyperbolic. It can also solve problems involving trigonometric functions like sin, cos, tan, and cot, as well as products and powers of these functions.

The AI can assist with integration using partial fractions, reduction formulae, and techniques like integration by parts. It can also help with double and triple integrals, numerical integration methods like the trapezoidal rule and Simpson's rule, and finding areas under and between curves.

Moreso, the AI can calculate volumes of solids of revolution, centroids of shapes, and solve first-order differential equations using techniques like separation of variables. With the ability to handle a wide range of integral calculus problems, this AI solver can be a valuable tool for students and professionals working in mathematics and related fields.

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1. $\int x^2 dx$

2. $\int 5x dx$

3. $\int \frac{1}{x} dx$

4. $\int e^x dx$

5. $\int \frac{1}{x^2} dx$

6. $\int \sin(x) dx$

7. $\int \cos(x) dx$

8. $\int \sin(2x) dx$

9. $\int \cos(2x) dx$

10. $\int \tan(3x) dx$

11. $\int \cot(2x) dx$

12. $\int x^3 dx$

13. $\int \frac{1}{x^3} dx$

14. $\int e^{2x} dx$

15. $\int \sin^2(x) dx$

16. $\int \cos^2(x) dx$

17. $\int \sin(x) \cos(x) dx$

18. $\int \sin(3x) dx$

19. $\int \cos(4x) dx$

20. $\int \sin(2x) \cos(2x) dx$

21. $\int \sin(\theta) d\theta$

22. $\int \cos(\theta) d\theta$

23. $\int \sin^2(\theta) d\theta$

24. $\int \cos^2(\theta) d\theta$

25. $\int \sin(\theta) \cos(\theta) d\theta$

26. $\int \tan(\theta) d\theta$

27. $\int \cot(\theta) d\theta$

28. $\int \sin(3\theta) d\theta$

29. $\int \cos(4\theta) d\theta$

30. $\int \sin(2\theta) \cos(2\theta) d\theta$

31. $\int \sinh(x) dx$

32. $\int \cosh(x) dx$

33. $\int \sinh(2x) dx$

34. $\int \cosh(3x) dx$

35. $\int \sinh(x) \cosh(x) dx$

36. $\int \sinh(\theta) d\theta$

37. $\int \cosh(\theta) d\theta$

38. $\int \sinh^2(\theta) d\theta$

39. $\int \cosh^2(\theta) d\theta$

40. $\int \sinh(\theta) \cosh(\theta) d\theta$

41. $\int e^{3x} dx$

42. $\int 2x^2 dx$

43. $\int \frac{1}{x^4} dx$

44. $\int e^{-x} dx$

45. $\int \cos(3x) dx$

46. $\int \tan^2(x) dx$

47. $\int \cot^3(x) dx$

48. $\int x^4 dx$

49. $\int \sin^3(x) dx$

50. $\int \cos(5x) dx$

Integral calculus
Standard integration
The process of integration
The general solution of integrals of the
form ax n
Standard integrals
Definite integrals
Integration using algebraic substitutions
Change of limits
Integration using trigonometric and hyperbolic
substitutions
Worked problems on integration of sin 2 x,
cos2 x, tan 2 x and cot 2 x
Worked problems on integration of powers
of sines and cosines
Worked problems on integration of
products of sines and cosines
Worked problems on integration using the
sin θ substitution
Worked problems on integration using the
tan θ substitution
Worked problems on integration using the
sinh θ substitution
Worked problems on integration using the
cosh θ substitution
Integration using partial fractions
Worked problems on integration using
partial fractions with linear factors
Worked problems on integration using
partial fractions with repeated linear factors
Worked problems on integration using
partial fractions with quadratic factors
The t = tan θ/2 substitution
Integration by parts
Reduction formulae
Using reduction formulae for integrals of
the form ∫ x n ex dx
Using reduction formulae for integrals of
the form ∫ x n cos x dx and ∫ x n sin x dx
Using reduction formulae for integrals of
the form ∫ sinn x dx and ∫ cosn x dx
Double and triple integrals
Numerical integration
The trapezoidal rule
The mid-ordinate rule
Simpson’s rule
Areas under and between curves
Area under a curve
The area between curves
Mean and root mean square values
Mean or average values
Root mean square values
Volumes of solids of revolution
Centroids of simple shapes
Solution of first-order differential equations
by separation of variables
Family of curves
Differential equations

Want to Learn More? Use Calulus AI

The AI Calculus Solver for integral calculus is equipped to handle:

Basic Integration: Polynomial, trigonometric, exponential, and logarithmic functions.

Definite and Indefinite Integrals: Solving for exact values or functions with constants of integration.

Techniques of Integration:

Substitution method.

Integration by parts.

Integration using partial fractions.

Trigonometric substitution.

Applications of Integrals:

Area under a curve.

Volume of solids of revolution (using the disc or shell method).

Accumulated change.

Worked Examples of Integral Calculus

1. Indefinite Integral of a Trigonometric Function

Problem:
Evaluate $ \int \sin^2(x) \, dx $.

Solution

1. Use the trigonometric identity:
\[
\sin^2(x) = \frac{1 – \cos(2x)}{2}.
\]
So,
\[
\int \sin^2(x) \, dx = \int \frac{1 – \cos(2x)}{2} \, dx.
\]

2. Split the integral:
\[
\int \sin^2(x) \, dx = \frac{1}{2} \int 1 \, dx – \frac{1}{2} \int \cos(2x) \, dx.
\]

3. Solve each term:
– $ \frac{1}{2} \int 1 \, dx = \frac{x}{2} $.
– For $ \frac{1}{2} \int \cos(2x) \, dx $, use substitution: Let $ u = 2x $, so $ du = 2 \, dx $.
\[
\int \cos(2x) \, dx = \frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) = \frac{1}{2} \sin(2x).
\]

4. Combine results:
\[
\int \sin^2(x) \, dx = \frac{x}{2} – \frac{\sin(2x)}{4} + C.
\]

Answer:
\[
\int \sin^2(x) \, dx = \frac{x}{2} – \frac{\sin(2x)}{4} + C.
\]

2. Definite Integral of an Exponential Function

Problem
Evaluate $ \int_0^1 e^{2x} \, dx $.

Solution
1. Identify the rule: Use the formula $ \int e^{ax} \, dx = \frac{e^{ax}}{a} + C $.
2. Solve the indefinite integral:
\[
\int e^{2x} \, dx = \frac{e^{2x}}{2}.
\]
3. Apply the limits (from 0 to 1):
\[
\int_0^1 e^{2x} \, dx = \left[ \frac{e^{2x}}{2} \right]_0^1.
\]
4. Evaluate:
\[
\int_0^1 e^{2x} \, dx = \frac{e^2}{2} – \frac{e^0}{2} = \frac{e^2}{2} – \frac{1}{2}.
\]

Answer:
\[
\int_0^1 e^{2x} \, dx = \frac{e^2 – 1}{2}.
\]

3. Integration by Substitution

Problem
Evaluate $ \int x \sqrt{1 + x^2} \, dx $.

Solution
1. Use substitution: Let $ u = 1 + x^2 $, so $ du = 2x \, dx $.
2. Rewrite the integral:
\[
\int x \sqrt{1 + x^2} \, dx = \frac{1}{2} \int \sqrt{u} \, du.
\]
3. Solve $ \int \sqrt{u} \, du $:
\[
\int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}.
\]
4. Back-substitute $ u = 1 + x^2 $:
\[
\frac{1}{2} \cdot \frac{2}{3} (1 + x^2)^{3/2} = \frac{1}{3} (1 + x^2)^{3/2} + C.
\]

Answer:
\[
\int x \sqrt{1 + x^2} \, dx = \frac{1}{3} (1 + x^2)^{3/2} + C.
\]

4. Integration by Parts

Problem
Evaluate $ \int x e^x \, dx $.

Solution
1. Use the formula for integration by parts:
\[
\int u \, dv = uv – \int v \, du.
\]
Let $ u = x $ and $ dv = e^x dx $.
2. Find $ du $ and $ v $:
– $ du = dx $,
– $ v = \int e^x dx = e^x $.
3. Substitute into the formula:
\[
\int x e^x \, dx = x e^x – \int e^x \, dx.
\]
4. Solve $ \int e^x \, dx $:
\[
\int x e^x \, dx = x e^x – e^x + C.
\]
5. Factorize:
\[
\int x e^x \, dx = e^x (x – 1) + C.
\]

Answer
\[
\int x e^x \, dx = e^x (x – 1) + C.
\]

5. Area Under a Curve

Problem
Find the area under $ y = x^2 $ between $ x = 0 $ and $ x = 3 $.

Solution
1. Set up the integral:
\[
\int_0^3 x^2 \, dx.
\]
2. Solve the indefinite integral:
\[
\int x^2 \, dx = \frac{x^3}{3}.
\]
3. Apply the limits:
\[
\int_0^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^3.
\]
4. Evaluate:
\[
\int_0^3 x^2 \, dx = \frac{3^3}{3} – \frac{0^3}{3} = \frac{27}{3} – 0 = 9.
\]

Answer
The area under the curve is $ 9 $ square units.

6. Volume of a Solid of Revolution

Problem
Find the volume of the solid obtained by rotating $ y = x^2 $ about the x-axis between $ x = 0 $ and $ x = 2 $.

Solution
1. Use the formula for volume of revolution:
\[
V = \pi \int_a^b [f(x)]^2 \, dx.
\]
2. Substitute $ f(x) = x^2 $:
\[
V = \pi \int_0^2 (x^2)^2 \, dx = \pi \int_0^2 x^4 \, dx.
\]
3. Solve the indefinite integral:
\[
\int x^4 \, dx = \frac{x^5}{5}.
\]
4. Apply the limits:
\[
V = \pi \left[ \frac{x^5}{5} \right]_0^2 = \pi \left( \frac{2^5}{5} – \frac{0^5}{5} \right).
\]
5. Simplify:
\[
V = \pi \left( \frac{32}{5} –