Update app.py

app.py

@@ -6,28 +6,29 @@ def main():
     st.title("Integration by Parts - Step-by-Step Solver")
     st.markdown(
         """
-        Integration by Parts formula:
-        \
+        ## Formula for Integration by Parts
+\[
         \\int u \\cdot v' \\, dx = u \\cdot v - \\int v \\cdot u' \\, dx
-        \
+\]
         """
     )
 
     # Input section
-    st.header("Input Function")
+    st.header("Input the Function")
     function_input = st.text_input(
-        "Enter the integrand (e.g., x * exp(x) or ln(x)):", "x * exp(x)"
+        "Enter the integrand (e.g., x * exp(x), ln(x), x^2 * cos(x)):",
+        "x * exp(x)",
     )
     u_input = st.text_input(
-        "Choose 'u' (e.g., x for x * exp(x)):",
-        "x"
+        "Choose 'u' (e.g., x for x * exp(x) or ln(x)):", "x"
     )
 
-    # Process inputs
     if st.button("Solve"):
         try:
+            # Define the variable
             x = symbols("x")
-            # Parse the integrand and chosen `u`
+            
+            # Parse inputs
             integrand = eval(function_input)
             u = eval(u_input)
             dv = integrand / u
@@ -36,18 +37,35 @@ def main():
             du = diff(u, x)
             v = integrate(dv, x)
 
-            # Apply the formula
-            result = u * v - integrate(v * du, x)
+            # Compute the result
+            first_term = u * v
+            second_term = integrate(v * du, x)
+            result = first_term - second_term
 
             # Display steps
             st.subheader("Step-by-Step Solution")
+            
+            st.markdown("### 1. Choose u and v':")
             st.latex(f"u = {latex(u)}, \\quad v' = {latex(dv)}")
-            st.latex(f"du = {latex(du)}, \\quad v = \\int {latex(dv)} \\, dx = {latex(v)}")
-            st.markdown("Substituting into the formula:")
+            
+            st.markdown("### 2. Compute du and v:")
+            st.latex(f"du = \\frac{{d}}{{dx}}[{latex(u)}] = {latex(du)}")
+            st.latex(f"v = \\int {latex(dv)} \\, dx = {latex(v)}")
+            
+            st.markdown("### 3. Apply the Integration by Parts Formula:")
             st.latex(
-                f"\\int {latex(integrand)} \\, dx = {latex(u)} \\cdot {latex(v)} - \\int {latex(v)} \\cdot {latex(du)} \\, dx"
+                f"\\int {latex(integrand)} \\, dx = u \\cdot v - \\int v \\cdot du"
             )
-            st.latex(f"= {latex(result)} + C")
+            st.latex(
+                f"= ({latex(u)})({latex(v)}) - \\int ({latex(v)})({latex(du)}) \\, dx"
+            )
+            
+            st.markdown("### 4. Evaluate the Terms:")
+            st.latex(f"First Term = {latex(first_term)}")
+            st.latex(f"Second Term = \\int {latex(v * du)} \\, dx = {latex(second_term)}")
+            
+            st.markdown("### 5. Combine Results:")
+            st.latex(f"\\int {latex(integrand)} \\, dx = {latex(result)} + C")
 
             # Final output
             st.subheader("Final Answer")