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Manual” Integration with Steps:

The calculator finds an antiderivative in a comprehensible way. Note that due to some simplifications, it might only be valid for parts of the function:

âˆ«(ln(y + vs)) âˆ«(ln(y – vs) – 2 + 53) + C 5y

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01:16

There are often multiple ways of computing an antiderivative. For $\int \frac{1}{x \ln \sqrt{x}} d x,$ first use the substitution $u=\ln \sqrt{x}$ to find the indefinite integral $2 \ln |\ln \sqrt{x}|+c .$ Then rewrite $\ln \sqrt{x}$ and use the substitution $u=\ln x$ to find the indefinite integral 2 In $|\ln x|+c .$ Show that these two answers are equivalent.

03:03

Evaluate using Integration by Parts as a first step.$$\int \ln \left(x^{2}+1\right) d x$$

02:38

In Exercises $53-56,$ evaluate using Integration by Parts as a first step.$$\int \ln \left(x^{2}+1\right) d x$$

01:39

Use integration by parts to evaluate the integral:âˆ«ln(x)â‹…âˆšx^3dx=

01:51

Guess an antiderivative for the integrand function. Validate your guess by differentiation, and then evaluate the given definite integral. (Hint: Keep the Chain Rule in mind when trying to guess an antiderivative. You will learn how to find such antiderivatives in the next section.)$$\int_{1}^{2} \frac{\ln x}{x} d x$$

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