<Dl> <Dd> ∫ 0 ∞ x e − a x cos ⁡ b x d x = a 2 − b 2 (a 2 + b 2) 2 (a> 0) (\ displaystyle \ int _ (0) ^ (\ infty) xe ^ (- ax) \ cos bx \, dx = (\ frac (a ^ (2) - b ^ (2)) ((a ^ (2) + b ^ (2)) ^ (2))) \ quad (a> 0)) </Dd> </Dl> <Dd> ∫ 0 ∞ x e − a x cos ⁡ b x d x = a 2 − b 2 (a 2 + b 2) 2 (a> 0) (\ displaystyle \ int _ (0) ^ (\ infty) xe ^ (- ax) \ cos bx \, dx = (\ frac (a ^ (2) - b ^ (2)) ((a ^ (2) + b ^ (2)) ^ (2))) \ quad (a> 0)) </Dd> <Dl> <Dd> ∫ 0 2 π e x cos ⁡ θ d θ = 2 π I 0 (x) (\ displaystyle \ int _ (0) ^ (2 \ pi) e ^ (x \ cos \ theta) d \ theta = 2 \ pi I_ (0) (x)) (I is the modified Bessel function of the first kind) </Dd> </Dl> <Dd> ∫ 0 2 π e x cos ⁡ θ d θ = 2 π I 0 (x) (\ displaystyle \ int _ (0) ^ (2 \ pi) e ^ (x \ cos \ theta) d \ theta = 2 \ pi I_ (0) (x)) (I is the modified Bessel function of the first kind) </Dd>

Integrals involving exponential and sine and cosine functions