To get the best result from integral calculus playground, use the following guides to correctly form your calculus  expression

The standard integration

1. Find the integral of 5x^2 + 3x + 7 with respect to x.
2. Compute the indefinite integral of \frac{1}{x} + \sqrt{x}.
3. Determine the integral of e^{2x} with respect to x.
4. Calculate the antiderivative of \cos(x) - \sin(x).
5. Find the integral of \frac{3x^2 + 2x - 1}{x^2} with respect to x.
6. Compute the indefinite integral of 2\sin(x) + 3\cos(x).
7. Determine the integral of \frac{x^3}{3} + 4x^2 - 2 with respect to x.
8. Calculate the antiderivative of \frac{1}{x} + \frac{1}{x^2}.
9. Find the integral of e^{3x} - \sin(2x) with respect to x.
10. Compute the indefinite integral of 5\sqrt{x} - \frac{1}{x}.

can be written correctly as

1. 5x^2 + 3x + 7
2. 1/x + sqrt(x)
3. e^(2x)
4. cos(x) - sin(x)
5. (3x^2 + 2x - 1)/(x^2)
6. 2sin(x) + 3cos(x)
7. (x^3/3) + 4x^2 - 2
8. 1/x + 1/x^2
9. e^(3x) - sin(2x)
10. 5sqrt(x) - 1/x

The integration of trigonometric functions

1. Find the integral of \sin(x) with respect to x.
2. Integrate \cos(3x) with respect to x.
3. Determine the integral of \tan^2(x) with respect to x.
4. Find the antiderivative of \sec(2x) \tan(2x) with respect to x.
5. Calculate the integral of \csc(x) \cot(x) with respect to x.
6. Determine the antiderivative of \sin(2x) \cos(2x) with respect to x.
7. Integrate \tan(x) \sec^2(x) with respect to x.
8. Find the integral of \sin^2(x) \cos(x) with respect to x.
9. Calculate the antiderivative of \cos(4x) \sin(4x) with respect to x.
10. Determine the integral of \cot(x) \csc^2(x) with respect to x.

can be written into the playground box as

1. sin(x)
2. cos(3x)
3. tan^2(x)
4. sec(2x) tan(2x)
5. csc(x) cot(x)
6. sin(2x) cos(2x)
7. tan(x) sec^2(x)
8. sin^2(x) cos(x)
9. cos(4x) sin(4x)
10. cot(x) csc^2(x)

About Integral Calculus Playground

The "Integral Calculus Playground" is a sophisticated educational tool designed to help users practice and enhance their understanding of integral calculus. This platform covers a wide range of topics, including standard integration techniques, definite integrals, integration using algebraic and trigonometric substitutions, change of limits, and more.

Users can explore worked problems on integrating functions such as sin 2x, cos 2x, tan 2x, and cot 2x, as well as powers and products of sines and cosines. The platform also provides practice on integration using substitutions such as sin θ, tan θ, sinh θ, and cosh θ, along with partial fractions with linear and quadratic factors. Additionally, users can learn about the t = tan θ/2 substitution, integration by parts, and reduction formulae.

With its comprehensive coverage of integral calculus concepts and interactive exercises, the "Integral Calculus Playground" is a valuable resource for students and educators alike seeking to improve their skills in this area of mathematics.

1. Find the integral of 3x^4.

Solution:

\int 3x^4 dx = 3 \int x^4 dx = 3 * \frac{x^5}{5} + C = \frac{3x^5}{5} + C

2. Evaluate the integral \int (2x^3 + 5x^2 - 4) dx.

Solution:

\int (2x^3 + 5x^2 - 4) dx = \int 2x^3 dx + \int 5x^2 dx - \int 4 dx = \frac{2x^4}{4} + \frac{5x^3}{3} - 4x + C = \frac{1}{2}x^4 + \frac{5}{3}x^3 - 4x + C

3. Solve the definite integral \int_{0}^{2} (x^2 + 3x - 1) dx.

Solution:

\int_{0}^{2} (x^2 + 3x - 1) dx = [\frac{x^3}{3} + \frac{3x^2}{2} - x]_{0}^{2} = \frac{8}{3} + 6 - 2 - 0 = \frac{14}{3}

4. Integrate \int \frac{1}{x} dx.

Solution:

\int \frac{1}{x} dx = \ln|x| + C

5. Find the integral of \sin^2(x).

Solution:

\int \sin^2(x) dx = \int \frac{1 - \cos(2x)}{2} dx = \frac{1}{2}x - \frac{\sin(2x)}{4} + C

6. Evaluate \int x\cos(x) dx.

Solution:

Integrate by parts: let  u = x and dv = cos(x) dx

Then, du = dx and v = \sin(x)

\int x\cos(x) dx = x\sin(x) - \int \sin(x) dx = x\sin(x) + \cos(x) + C

7. Compute \int \frac{x}{x^2 + 4} dx.

Solution:

Let  u = x^2 + 4, then du = 2x dx

\int \frac{x}{x^2 + 4} dx = \frac{1}{2} \ln|x^2 + 4| + C

8. Solve the integral \int \frac{2x + 3}{x^2 - 4} dx.

Solution:

Use partial fractions: \frac{2x + 3}{x^2 - 4} = \frac{A}{x+2} + \frac{B}{x-2}

Solving for A and B, we get A = 1 and B = 1

\int \frac{2x + 3}{x^2 - 4} dx = \int \frac{1}{x+2} dx + \int \frac{1}{x-2} dx = \ln|x+2| + \ln|x-2| + C

9. Evaluate \int \frac{1}{1 + \cos(x)} dx.

Solution:

Multiply and divide by (1 - \cos(x)) to simplify the integral:

\int \frac{1}{1 + \cos(x)} dx = \int \frac{1 - \cos(x)}{\sin^2(x)} dx = -\cot(x) + C

10. Compute the integral \int \sin(2x) dx.

Solution:

Use the double angle formula: \sin(2x) = 2\sin(x)\cos(x)

\int \sin(2x) dx = -\frac{1}{2}\cos(2x) + C = -\cos(2x) + C

Questions

1. \frac{d}{dx}(f(x)g(x))

2. \frac{d}{dx}(3x^2 \cdot \sin(x))

3. \frac{d}{dx}(e^x \cdot \cos(x))

4. \frac{d}{dx}(2x \cdot \ln(x))

5. \frac{d}{dx}(x^3 \cdot e^{2x})

6. \frac{d}{dx}(4x \cdot \tan(x))

7. \frac{d}{dx}(6x^2 \cdot \sqrt{x})

8. \frac{d}{dx}(\ln(x) \cdot e^x)

9. \frac{d}{dx}(2x^3 \cdot \sin(2x))

10. \frac{d}{dx}(x \cdot \cos(x))

11. \frac{d}{dx}(4x^2 \cdot e^x)

12. \frac{d}{dx}(e^{3x} \cdot \tan(x))

13. \frac{d}{dx}(5x \cdot \ln(2x))

14. \frac{d}{dx}(x^4 \cdot \cosh(x))

15. \frac{d}{dx}(e^{-x} \cdot \sin(x))

Answer:

1. g(x)+f(x)g'(x)

2. 6x\sin(x)+3x^2\cos(x)

3. e^x\cos(x)-e^x\sin(x)

4. 2\ln(x)+2

5. 3x^2e^{2x}+2x^3e^{2x}

6. 4\tan(x)+4x\sec^2(x)

7. 12x\sqrt{x}+3x^2\sqrt{x}

8. \frac{1}{x}e^x+\ln(x)e^x

9. 6x^2\sin(2x)+4x^3\cos(2x)

10. \cos(x)-x\sin(x)

11. 4x^2e^x+8xe^x

12. 3e^{3x}\tan(x)+e^{3x}\sec^2(x)

13. 5\ln(2x)+5 [\frac{1}{x}]

14. 4x^3\cosh(x)+x^4\sinh(x)

15. -e^{-x}\sin(x)-e^{-x}\cos(x)