Then it's Antiderivative (in general form) is F(x) + C, where C is an arbitrary constant.

f(x) = x² + x + 5

Solution: The general antiderivative of f(x) is

F(x) = (1/3)x³ + (1/2)x² + 5x + C.

Find the antiderivative of the trigonometric function.

f(x) = cos(x) + sin(x)

Solution: The derivative of cos(x) is -sin(x) and sin(x) is cos(x).

So the antiderivative of f(x) is the opposite.

Therefore, F(x) = sin(x) + [-cos(x)] = sin(x) - cos(x).

Taking antiderivative of a polynomial function is like using Power Rule, but instead, you add one to the exponent and divide by the new exponent.

For the trigonometric functions, they alternate with negative sign.

Find f(x) if f''(x) = (x)^1/2

Now you are ready to do the integral. For definite integral, there exists both upper limit* and lower limit* as real numbers.

*where upper limit is a and lower limit is b.

∫₀¹ x dx

Solution: First, antiderivative of x is (1/2)x².

So, ∫₀¹ x dx = (1/2)x² l₀¹ = (1/2)(1)² - (1/2)(0) = 1/2.

Evaluate the integral.

∫₀¹ cos(x) dx

Solution: ∫₀¹ cos(x) dx = sin(x) l₀¹ = sin(1) - sin(0) = sin(1).

Prove that ∫_b^a x dx = (b² - a²)/2.