Integration by Substitution/Examples/Cosine over Square of 1 + Sine

Example of Use of Integration by Substitution

$\ds \int \dfrac {\cos x} {\paren {1 + \sin x}^2} \rd x = -\dfrac 1 {1 + \sin x} + C$

Proof

\(\ds u\)

\(=\)

\(\ds \sin x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds \cos x\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {\cos x} {\paren {1 + \sin x}^2} \rd x\)

\(=\)

\(\ds \int \dfrac 1 {\paren {1 + u}^2} \rd u\)

Primitive of Composite Function: Corollary

\(\ds \)

\(=\)

\(\ds -\dfrac 1 {1 + u} + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds -\dfrac 1 {1 + \sin x} + C\)

substituting for $u$ and simplifying

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $7$.