Integrate Using u-Substitution integral of sin(x)sin(cos(x)) with respect to x
This problem is asking for the integration of the given function sin(x)sin(cos(x)) with respect to the variable x using the u-substitution method. The u-substitution technique involves changing the variable of integration from x to a new variable u, which is a function of x. This is done in the hopes of simplifying the integral or turning it into a more recognizable form that can be easily integrated. For this specific integral, it may require identifying the appropriate substitution to make for u, which is a function that, when substituted, will allow the integral to be expressed in terms of u and du rather than x and dx. Once the expression is integrated in terms of u, the final step would be to substitute back in terms of x to find the solution.
$\int sin \left(\right. x \left.\right) sin \left(\right. cos \left(\right. x \left.\right) \left.\right) d x$
Answer
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:
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 $u$.
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 $u$.
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.
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 $\sin(u)$.
Back-Substitution: After integrating with respect to $u$, replace $u$ with the original expression in terms of $x$ to return to the original variable.
Simplification: Simplify the resulting expression if necessary. This may include combining like terms or applying algebraic identities.
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.
