Antiderivatives and indefinite integrals

Integrate to obtain F(x) with an unknown constant. Substitute the given x and F value to determine the constant. Then evaluate the requested function value.

Differentiate the proposed F, including the outer exponent and the inner linear derivative. Match the resulting coefficient to that of f.

The two antiderivatives have the same derivative. Their difference therefore has derivative zero and is constant on the interval. Determine that difference at the point where both values are known.

Rewrite the fraction as a negative power of x, apply the power rule and differentiate to check. The logarithm rule applies to exponent −1, not to every fraction.

Integrate each term separately, retaining its sign. The antiderivative of a constant term depends on x. One arbitrary constant is sufficient for the complete sum.

Apply the power rule for integration: increase the exponent by one and divide the coefficient by the new exponent. Add an arbitrary constant. Differentiate to check.

Choose the corresponding sine, cosine or exponential antiderivative. Account for the derivative sign and divide by the coefficient of x in the linear argument. Check using the chain rule.

Recall which function has derivative 1/x. Retain the constant numerator factor and add an arbitrary constant. The positive domain allows the logarithm to be written without absolute-value signs.