Problem 43 - Essential Calculus

Question

Find or evaluate the integral using substitution first, then using integration by parts. $$ \int \cos (\ln x) d x $$

Step-by-Step Solution

Answer

The result of the integral \( \int \cos (\ln x) dx \) using both methods, substitution and integration by parts, is \( x\ln{x} - x + C \).

Step 1: Identify a suitable substitution

Identify that \( u = \ln x \) is a suitable substitution. Use this to simplify the integral. We must also calculate the derivative of \( u \), which is \( du = (1/x) dx \). Multiplied both sides by \( x \), we get \( dx = x du \). This substitution changes the integral into a format that we can evaluate.

Step 2: Compute the integral with substitution

Replace \( \ln x \) and \( dx \) in the equation with the substitution results to get \( \int cos(u) \cdot x du \). This integral can be solved using integration by parts.

Step 3: Integration by parts

For the integration by parts, let \( v = x \) and \( du = cos(u) du \). Then, \( dv = dx \) and \( u = \ln{x} \). We can use integration by parts formula \( \int udv = uv - \int vdu \) substituting the u, v, du, and dv accordingly. The integral thus becomes \( \int cos(u) \cdot x du = x\ln{x} - \int x du \).

Step 4: Further simplification and final integral

The next integral to be solved is \( \int x du \). This can be simplified to \( \int x (1/x) dx = \int dx \). Solving this integral gives \( x \). Therefore, the final integral becomes: \( x\ln{x} - x + C \) where C is the integration constant.