Integrals!

Calculus Level 3

If ∫ x e sin ⁡ x d x = F ( x ) + c \int x e^{\sin x} \, dx = F(x) + c , and ∫ 0 10 x 3 e sin ⁡ ( x 2 ) d x = k ( F ( a ) − F ( b ) ) \displaystyle \int_0^{10} x^3 e^{\sin (x^2)} \, dx = k (F(a) - F(b)) . Compute k a + b ka + b .

This question is a part of the set Calculus .


The answer is 50.

Sahil Bansal by Sahil Bansal , 21, India

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1 solution

Sahil Bansal
Mar 19, 2016

Making the substitution x 2 = t x^{2}=t in the required integral,

∫ 0 10 x 3 e s i n x 2 d x = ∫ 0 100 t 2 e s i n t d t \displaystyle \int_0^{10} x^3 e^{sin x^{2}}dx = \displaystyle \int_0^{100} \frac{t}{2} e^{sin t}dt

Hence required integral is 1 2 [ F ( 100 ) − F ( 0 ) ] \frac{1}{2} [F(100)-F(0)]

So, k = 1 2 =\frac{1}{2} , a = 100 =100 , b = 0 =0 . Hence k a + b = 50 ka+b=50

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