Integrate Using u-Substitution integral of sin(x)sin(cos(x)) with respect to x

Solution:

Step 1

Assign $u=\cos (x)$. Consequently, $du=-\sin (x)dx$, which implies $-\frac{1}{\sin (x)}du=dx$. Substitute $u$ and $du$ into the integral.

Step 1.1

Set $u=\cos (x)$ and calculate $\frac{du}{dx}$.

Step 1.1.1

Take the derivative of $\cos (x)$: $\frac{d}{dx}[\cos (x)]$.

Step 1.1.2

The derivative of $\cos (x)$ with respect to $x$ is $-\sin (x)$: $-\sin (x)$.

Step 1.2

Express the integral in terms of $u$ and $du$: $\int -1\sin (u)du$.

Step 2

Extract the constant $-1$ from the integral as it is independent of $u$: $-\int \sin (u)du$.

Step 3

Integrate $\sin (u)$ with respect to $u$ to obtain $-\cos (u)$: $-(-\cos (u)+C)$.

Step 4

Proceed to simplify the expression.

Step 4.1

Simplify: $--\cos (u)+C$.

Step 4.2

Further simplification.

Step 4.2.1

Multiply $-1$ by $-1$: $1\cos (u)+C$.

Step 4.2.2

Multiply $\cos (u)$ by $1$: $\cos (u)+C$.

Step 5

Substitute back the original variable: Replace $u$ with $\cos (x)$ to get $\cos (\cos (x))+C$.

Knowledge Notes:

The process of solving an integral using u-substitution involves several key steps and knowledge points:

1. Choosing the Substitution: Identify a part of the integral that when differentiated, appears elsewhere in the integral. This is often a function inside another function, making it a good candidate for $u$.

2. Differentiating $u$: Compute $\frac{du}{dx}$ to find the relationship between $dx$ and $du$. This step is crucial to rewrite the integral entirely in terms of $u$.

3. Rewriting the Integral: Substitute $u$ and $du$ into the integral to replace all instances of $x$ and $dx$. This often simplifies the integral into a standard form that is easier to solve.

4. Integrating with Respect to $u$: Perform the integration with the new variable $u$. This might involve recognizing standard integral forms, such as the integral of $\sin (u)$.

5. Back-Substitution: After integrating with respect to $u$, replace $u$ with the original expression in terms of $x$ to return to the original variable.

6. Simplification: Simplify the resulting expression if necessary. This may include combining like terms or applying algebraic identities.

7. Adding the Constant of Integration: Since indefinite integrals represent a family of functions differing by a constant, always include the constant of integration $C$ in the final answer.

In this problem, the integral of $\sin (x)\sin (\cos (x))$ with respect to $x$ is transformed by recognizing $\cos (x)$ as a suitable substitution for $u$. The process involves differentiation, substitution, integration, and algebraic simplification to arrive at the final result.